{"id":183,"date":"2019-08-20T17:03:54","date_gmt":"2019-08-20T21:03:54","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/solving-systems-with-cramers-rule\/"},"modified":"2022-06-01T10:39:37","modified_gmt":"2022-06-01T14:39:37","slug":"solving-systems-with-cramers-rule","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/solving-systems-with-cramers-rule\/","title":{"raw":"Solving Systems with Cramer&#8217;s Rule","rendered":"Solving Systems with Cramer&#8217;s Rule"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section, you will:\n<ul>\n \t<li>Evaluate \u20092 \u00d7 2\u2009 determinants.<\/li>\n \t<li>Use Cramer\u2019s Rule to solve a system of equations in two variables.<\/li>\n \t<li>Evaluate \u20093 \u00d7 3\u2009 determinants.<\/li>\n \t<li>Use Cramer\u2019s Rule to solve a system of three equations in three variables.<\/li>\n \t<li>Know the properties of determinants.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1705215\">We have learned how to solve systems of equations in two variables and three variables, and by multiple methods: substitution, addition, Gaussian elimination, using the inverse of a matrix, and graphing. Some of these methods are easier to apply than others and are more appropriate in certain situations. In this section, we will study two more strategies for solving systems of equations.<\/p>\n\n<div id=\"fs-id1423475\" class=\"bc-section section\">\n<h3>Evaluating the Determinant of a 2\u00d72 Matrix<\/h3>\n<p id=\"fs-id1331675\">A determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a <span class=\"no-emphasis\">square matrix<\/span> to determine whether there is a solution to the system of equations. Perhaps one of the more interesting applications, however, is their use in cryptography. Secure signals or messages are sometimes sent encoded in a matrix. The data can only be decrypted with an <span class=\"no-emphasis\">invertible matrix<\/span> and the determinant. For our purposes, we focus on the determinant as an indication of the invertibility of the matrix. Calculating the determinant of a matrix involves following the specific patterns that are outlined in this section.<\/p>\n\n<div id=\"fs-id1305022\">\n<h3>Find the Determinant of a 2 \u00d7 2 Matrix<\/h3>\n<p id=\"fs-id1376906\">The determinant of a[latex]\\,2\\text{ }\u00d7\\text{ }2\\,[\/latex]matrix, given<\/p>\n\n<div id=\"fs-id1355183\" class=\"unnumbered aligncenter\">[latex]A=\\left[\\begin{array}{cc}a&amp; b\\\\ c&amp; d\\end{array}\\right][\/latex]<\/div>\n<p id=\"fs-id1644238\">is defined as<\/p>\n<span id=\"fs-id1347161\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154150\/CNX_Precalc_Figure_09_08_001.jpg\" alt=\"\"><\/span>\n<p id=\"fs-id1531976\">Notice the change in notation. There are several ways to indicate the determinant, including[latex]\\,\\mathrm{det}\\left(A\\right)\\,[\/latex]and replacing the brackets in a matrix with straight lines,[latex]\\,|A|.[\/latex]<\/p>\n\n<\/div>\n<div id=\"Example_09_08_01\" class=\"textbox examples\">\n<div id=\"fs-id1046032\">\n<div id=\"fs-id1426119\">\n<h3>Finding the Determinant of a 2 \u00d7 2 Matrix<\/h3>\nFind the determinant of the given matrix.\n<div id=\"fs-id1707032\" class=\"unnumbered aligncenter\">[latex]A=\\left[\\begin{array}{cc}5&amp; 2\\\\ -6&amp; 3\\end{array}\\right][\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"1408516\"]Show Solution[\/reveal-answer][hidden-answer a=\"1408516\"]\n<div id=\"fs-id1507367\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\mathrm{det}\\left(A\\right)=|\\begin{array}{cc}5&amp; 2\\\\ -6&amp; 3\\end{array}|\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=5\\left(3\\right)-\\left(-6\\right)\\left(2\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=27\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1420496\" class=\"bc-section section\">\n<h3>Using Cramer\u2019s Rule to Solve a System of Two Equations in Two Variables<\/h3>\n<p id=\"fs-id1322833\">We will now introduce a final method for solving systems of equations that uses determinants. Known as <span class=\"no-emphasis\">Cramer\u2019s Rule<\/span>, this technique dates back to the middle of the 18th century and is named for its innovator, the Swiss mathematician Gabriel Cramer (1704-1752), who introduced it in 1750 in Introduction \u00e0 l'Analyse des lignes Courbes alg\u00e9briques. Cramer\u2019s Rule is a viable and efficient method for finding solutions to systems with an arbitrary number of unknowns, provided that we have the same number of equations as unknowns.<\/p>\n<p id=\"fs-id1531826\">Cramer\u2019s Rule will give us the unique solution to a system of equations, if it exists. However, if the system has no solution or an infinite number of solutions, this will be indicated by a determinant of zero. To find out if the system is inconsistent or dependent, another method, such as elimination, will have to be used.<\/p>\n<p id=\"fs-id1393178\">To understand Cramer\u2019s Rule, let\u2019s look closely at how we solve systems of linear equations using basic row operations. Consider a system of two equations in two variables.<\/p>\n\n<div id=\"fs-id1279412\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}{a}_{1}x+{b}_{1}y={c}_{1}\\,\\,\\,\\,\\left(1\\right)\\\\ {a}_{2}x+{b}_{2}y={c}_{2}\\,\\,\\,\\,\\left(2\\right)\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1361346\">We eliminate one variable using row operations and solve for the other. Say that we wish to solve for[latex]\\,x.\\,[\/latex]If equation (2) is multiplied by the opposite of the coefficient of[latex]\\,y\\,[\/latex]in equation (1), equation (1) is multiplied by the coefficient of[latex]\\,y\\,[\/latex]in equation (2), and we add the two equations, the variable[latex]\\,y\\,[\/latex]will be eliminated.<\/p>\n\n<div id=\"fs-id1422407\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\underset{\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_}{\\begin{array}{llll}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill \\\\ \\,\\,\\,\\,{b}_{2}{a}_{1}x+{b}_{2}{b}_{1}y={b}_{2}{c}_{1}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Multiply }{R}_{1}\\text{ by }{b}_{2}\\hfill \\\\ -{b}_{1}{a}_{2}x-{b}_{1}{b}_{2}y=-{b}_{1}{c}_{2}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Multiply }{R}_{2}\\text{ by}-{b}_{1}\\hfill \\end{array}}\\hfill \\\\ \\,\\,\\,\\begin{array}{ll} {b}_{2}{a}_{1}x-{b}_{1}{a}_{2}x={b}_{2}{c}_{1}-{b}_{1}{c}_{2}\\hfill &amp; \\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1370413\">Now, solve for[latex]\\,x.[\/latex]<\/p>\n\n<div id=\"fs-id1460232\">[latex]\\begin{array}{l}\\,\\,\\,{b}_{2}{a}_{1}x-{b}_{1}{a}_{2}x={b}_{2}{c}_{1}-{b}_{1}{c}_{2}\\hfill \\\\ \\,\\,\\,x\\left({b}_{2}{a}_{1}-{b}_{1}{a}_{2}\\right)={b}_{2}{c}_{1}-{b}_{1}{c}_{2}\\hfill \\\\ \\text{ }x=\\frac{{b}_{2}{c}_{1}-{b}_{1}{c}_{2}}{{b}_{2}{a}_{1}-{b}_{1}{a}_{2}}=\\frac{\\left[\\begin{array}{cc}{c}_{1}&amp; {b}_{1}\\\\ {c}_{2}&amp; {b}_{2}\\end{array}\\right]}{\\left[\\begin{array}{cc}{a}_{1}&amp; {b}_{1}\\\\ {a}_{2}&amp; {b}_{2}\\end{array}\\right]}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1347808\">Similarly, to solve for[latex]\\,y,[\/latex]we will eliminate[latex]\\,x.[\/latex]<\/p>\n\n<div id=\"fs-id1286733\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\underset{\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_}{\\begin{array}{llll}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill \\\\ \\,\\,\\,\\,{a}_{2}{a}_{1}x+{a}_{2}{b}_{1}y={a}_{2}{c}_{1}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Multiply }{R}_{1}\\text{ by }{a}_{2}\\hfill \\\\ -{a}_{1}{a}_{2}x-{a}_{1}{b}_{2}y=-{a}_{1}{c}_{2}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Multiply }{R}_{2}\\text{ by}-{a}_{1}\\hfill \\end{array}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\begin{array}{ll}{a}_{2}{b}_{1}y-{a}_{1}{b}_{2}y={a}_{2}{c}_{1}-{a}_{1}{c}_{2}\\hfill &amp; \\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1647752\">Solving for[latex]\\,y\\,[\/latex]gives<\/p>\n\n<div id=\"fs-id1458502\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{a}_{2}{b}_{1}y-{a}_{1}{b}_{2}y={a}_{2}{c}_{1}-{a}_{1}{c}_{2}\\hfill \\\\ y\\left({a}_{2}{b}_{1}-{a}_{1}{b}_{2}\\right)={a}_{2}{c}_{1}-{a}_{1}{c}_{2}\\hfill \\\\ \\text{ }y=\\frac{{a}_{2}{c}_{1}-{a}_{1}{c}_{2}}{{a}_{2}{b}_{1}-{a}_{1}{b}_{2}}=\\frac{{a}_{1}{c}_{2}-{a}_{2}{c}_{1}}{{a}_{1}{b}_{2}-{a}_{2}{b}_{1}}=\\frac{|\\begin{array}{cc}{a}_{1}&amp; {c}_{1}\\\\ {a}_{2}&amp; {c}_{2}\\end{array}|}{|\\begin{array}{cc}{a}_{1}&amp; {b}_{1}\\\\ {a}_{2}&amp; {b}_{2}\\end{array}|}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1281751\">Notice that the denominator for both[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]is the determinant of the coefficient matrix.<\/p>\n<p id=\"fs-id1332312\">We can use these formulas to solve for[latex]\\,x\\,[\/latex]and[latex]\\,y,\\,[\/latex]but Cramer\u2019s Rule also introduces new notation:<\/p>\n\n<ul id=\"fs-id1279395\">\n \t<li>[latex]\\,\\,D:[\/latex]determinant of the coefficient matrix<\/li>\n \t<li>[latex]{D}_{x}:[\/latex]determinant of the numerator in the solution of[latex]x[\/latex]\n<div id=\"fs-id1972131\" class=\"unnumbered aligncenter\">[latex]x=\\frac{{D}_{x}}{D}[\/latex]<\/div><\/li>\n \t<li>[latex]{D}_{y}:[\/latex]determinant of the numerator in the solution of[latex]\\,y[\/latex]\n<div id=\"fs-id1294030\" class=\"unnumbered aligncenter\">[latex]y=\\frac{{D}_{y}}{D}[\/latex]<\/div><\/li>\n<\/ul>\n<p id=\"fs-id1422623\">The key to Cramer\u2019s Rule is replacing the variable column of interest with the constant column and calculating the determinants. We can then express[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]as a quotient of two determinants.<\/p>\n\n<div id=\"fs-id1357295\">\n<h3>Cramer\u2019s Rule for 2\u00d72 Systems<\/h3>\n<p id=\"fs-id1664070\">Cramer\u2019s Rule is a method that uses determinants to solve systems of equations that have the same number of equations as variables.<\/p>\n<p id=\"fs-id1660256\">Consider a system of two linear equations in two variables.<\/p>\n\n<div id=\"fs-id1700631\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}{a}_{1}x+{b}_{1}y={c}_{1}\\\\ {a}_{2}x+{b}_{2}y={c}_{2}\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1338021\">The solution using Cramer\u2019s Rule is given as<\/p>\n\n<div id=\"fs-id1375892\">[latex]x=\\frac{{D}_{x}}{D}=\\frac{|\\begin{array}{cc}{c}_{1}&amp; {b}_{1}\\\\ {c}_{2}&amp; {b}_{2}\\end{array}|}{|\\begin{array}{cc}{a}_{1}&amp; {b}_{1}\\\\ {a}_{2}&amp; {b}_{2}\\end{array}|},\\,\\,D\\ne 0;\\,\\,\\text{\u200b}\\text{\u200b}\\,y=\\frac{{D}_{y}}{D}=\\frac{|\\begin{array}{cc}{a}_{1}&amp; {c}_{1}\\\\ {a}_{2}&amp; {c}_{2}\\end{array}|}{|\\begin{array}{cc}{a}_{1}&amp; {b}_{1}\\\\ {a}_{2}&amp; {b}_{2}\\end{array}|},\\,\\,D\\ne 0.[\/latex]<\/div>\nIf we are solving for[latex]\\,x,\\,[\/latex]the[latex]\\,x\\,[\/latex]column is replaced with the constant column. If we are solving for[latex]\\,y,\\,[\/latex]the[latex]\\,y\\,[\/latex]column is replaced with the constant column.\n\n<\/div>\n<div id=\"Example_09_08_02\" class=\"textbox examples\">\n<div id=\"fs-id1455937\">\n<div id=\"fs-id1455939\">\n<h3>Using Cramer\u2019s Rule to Solve a 2 \u00d7 2 System<\/h3>\n<p id=\"fs-id1421342\">Solve the following[latex]\\,2\\text{ }\u00d7\\text{ }2\\,[\/latex]system using Cramer\u2019s Rule.<\/p>\n\n<div id=\"fs-id1536728\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}12x+3y=15\\\\ \\text{ }2x-3y=13\\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1459617\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1459617\"]\n<p id=\"fs-id1459617\">Solve for[latex]\\,x.[\/latex]<\/p>\n\n<div id=\"fs-id1536461\" class=\"unnumbered aligncenter\">[latex]x=\\frac{{D}_{x}}{D}=\\frac{|\\begin{array}{rr}\\hfill 15&amp; \\hfill 3\\\\ \\hfill 13&amp; \\hfill -3\\end{array}|}{|\\begin{array}{rr}\\hfill 12&amp; \\hfill 3\\\\ \\hfill 2&amp; \\hfill -3\\end{array}|}=\\frac{-45-39}{-36-6}=\\frac{-84}{-42}=2[\/latex]<\/div>\n<p id=\"fs-id1403542\">Solve for[latex]\\,y.[\/latex]<\/p>\n\n<div id=\"fs-id1697344\" class=\"unnumbered aligncenter\">[latex]y=\\frac{{D}_{y}}{D}=\\frac{|\\begin{array}{rr}\\hfill 12&amp; \\hfill 15\\\\ \\hfill 2&amp; \\hfill 13\\end{array}|}{|\\begin{array}{rr}\\hfill 12&amp; \\hfill 3\\\\ \\hfill 2&amp; \\hfill -3\\end{array}|}=\\frac{156-30}{-36-6}=-\\frac{126}{42}=-3[\/latex]<\/div>\n<p id=\"fs-id1406547\">The solution is[latex]\\,\\left(2,-3\\right).[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1034376\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_09_08_01\">\n<div id=\"fs-id1352281\">\n<p id=\"fs-id1352282\">Use Cramer\u2019s Rule to solve the 2 \u00d7 2 system of equations.<\/p>\n\n<div id=\"fs-id1647536\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{ }x+2y=-11\\hfill \\\\ -2x+y=-13\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1440640\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1440640\"]\n<p id=\"fs-id1440640\">[latex]\\left(3,-7\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1594718\" class=\"bc-section section\">\n<h3>Evaluating the Determinant of a 3 \u00d7 3 Matrix<\/h3>\n<p id=\"fs-id1673681\">Finding the determinant of a 2\u00d72 matrix is straightforward, but finding the determinant of a 3\u00d73 matrix is more complicated. One method is to augment the 3\u00d73 matrix with a repetition of the first two columns, giving a 3\u00d75 matrix. Then we calculate the sum of the products of entries <em>down<\/em> each of the three diagonals (upper left to lower right), and subtract the products of entries <em>up<\/em> each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.<\/p>\nFind the <span class=\"no-emphasis\">determinant<\/span> of the 3\u00d73 matrix.\n<div id=\"fs-id1505641\" class=\"unnumbered aligncenter\">[latex]A=\\left[\\begin{array}{ccc}{a}_{1}&amp; {b}_{1}&amp; {c}_{1}\\\\ {a}_{2}&amp; {b}_{2}&amp; {c}_{2}\\\\ {a}_{3}&amp; {b}_{3}&amp; {c}_{3}\\end{array}\\right][\/latex]<\/div>\n<ol id=\"fs-id1455932\" type=\"1\">\n \t<li>Augment[latex]\\,A\\,[\/latex]with the first two columns.\n<div id=\"fs-id1354963\" class=\"unnumbered aligncenter\">[latex]\\mathrm{det}\\left(A\\right)=|\\begin{array}{ccc}{a}_{1}&amp; {b}_{1}&amp; {c}_{1}\\\\ {a}_{2}&amp; {b}_{2}&amp; {c}_{2}\\\\ {a}_{3}&amp; {b}_{3}&amp; {c}_{3}\\end{array}\\,\\,\\,|\\,\\,\\,\\begin{array}{c}{a}_{1}\\\\ {a}_{2}\\\\ {a}_{3}\\end{array}\\,\\,\\,\\,\\begin{array}{c}{b}_{1}\\\\ {b}_{2}\\\\ {b}_{3}\\end{array}|[\/latex]<\/div><\/li>\n \t<li>From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.<\/li>\n \t<li>From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.<\/li>\n<\/ol>\n<span id=\"fs-id1536830\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154152\/CNX_Precalc_Figure_09_08_002.jpg\" alt=\"\"><\/span>\n<p id=\"fs-id1704961\">The algebra is as follows:<\/p>\n\n<div id=\"fs-id1704964\" class=\"unnumbered aligncenter\">[latex]|A|={a}_{1}{b}_{2}{c}_{3}+{b}_{1}{c}_{2}{a}_{3}+{c}_{1}{a}_{2}{b}_{3}-{a}_{3}{b}_{2}{c}_{1}-{b}_{3}{c}_{2}{a}_{1}-{c}_{3}{a}_{2}{b}_{1}[\/latex]<\/div>\n<div id=\"Example_09_08_03\" class=\"textbox examples\">\n<div id=\"fs-id1277069\">\n<div id=\"fs-id1277071\">\n<h3>Finding the Determinant of a 3 \u00d7 3 Matrix<\/h3>\n<p id=\"fs-id1529523\">Find the determinant of the 3 \u00d7 3 matrix given<\/p>\n\n<div id=\"fs-id1529527\" class=\"unnumbered aligncenter\">[latex]A=\\left[\\begin{array}{ccc}0&amp; 2&amp; 1\\\\ 3&amp; -1&amp; 1\\\\ 4&amp; 0&amp; 1\\end{array}\\right][\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"1570002\"]Show Solution[\/reveal-answer][hidden-answer a=\"1570002\"]\n\nAugment the matrix with the first two columns and then follow the formula. Thus,\n<div class=\"unnumbered\">[latex]\\begin{array}{l}|A|=|\\begin{array}{ccc}0&amp; 2&amp; 1\\\\ 3&amp; -1&amp; 1\\\\ 4&amp; 0&amp; 1\\end{array}\\,\\,|\\begin{array}{c}0\\\\ 3\\\\ \\,\\,4\\end{array}\\,\\,\\,\\,\\begin{array}{c}2\\\\ -1\\\\ 0\\end{array}|\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,=0\\left(-1\\right)\\left(1\\right)+2\\left(1\\right)\\left(4\\right)+1\\left(3\\right)\\left(0\\right)-4\\left(-1\\right)\\left(1\\right)-0\\left(1\\right)\\left(0\\right)-1\\left(3\\right)\\left(2\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,=0+8+0+4-0-6\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,=6\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1664049\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_09_08_02\">\n<div id=\"fs-id1608913\">\n<p id=\"fs-id1608914\">Find the determinant of the 3 \u00d7 3 matrix.<\/p>\n\n<div id=\"fs-id1608917\" class=\"unnumbered aligncenter\">[latex]\\mathrm{det}\\left(A\\right)=|\\begin{array}{ccc}1&amp; -3&amp; 7\\\\ 1&amp; 1&amp; 1\\\\ 1&amp; -2&amp; 3\\end{array}|[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1643971\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1643971\"]\n<p id=\"fs-id1643971\">[latex]-10[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1424491\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1445990\"><strong>Can we use the same method to find the determinant of a larger matrix?<\/strong><\/p>\n<p id=\"fs-id1445994\"><em>No, this method only works for[latex]\\,2\\text{ }\u00d7\\text{ }2\\,[\/latex]and[latex]\\,\\text{3}\\text{ }\u00d7\\text{ }3\\,[\/latex]matrices. For larger matrices it is best to use a graphing utility or computer software.<\/em><\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1420022\" class=\"bc-section section\">\n<h3>Using Cramer\u2019s Rule to Solve a System of Three Equations in Three Variables<\/h3>\n<p id=\"fs-id1433754\">Now that we can find the <span class=\"no-emphasis\">determinant<\/span> of a 3 \u00d7 3 matrix, we can apply <span class=\"no-emphasis\">Cramer\u2019s Rule<\/span> to solve a <span class=\"no-emphasis\">system of three equations in three variables<\/span>. Cramer\u2019s Rule is straightforward, following a pattern consistent with Cramer\u2019s Rule for 2 \u00d7 2 matrices. As the order of the matrix increases to 3 \u00d7 3, however, there are many more calculations required.<\/p>\n<p id=\"fs-id1433166\">When we calculate the determinant to be zero, Cramer\u2019s Rule gives no indication as to whether the system has no solution or an infinite number of solutions. To find out, we have to perform elimination on the system.<\/p>\n<p id=\"fs-id1518566\">Consider a 3 \u00d7 3 system of equations.<\/p>\n<span id=\"eip-id1435181\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154154\/9.8.1.jpg\" alt=\"\"><\/span>\n<div id=\"fs-id1452172\" class=\"unnumbered aligncenter\">[latex]x=\\frac{{D}_{x}}{D},y=\\frac{{D}_{y}}{D},z=\\frac{{D}_{z}}{D},D\\ne 0[\/latex]<\/div>\n<p id=\"fs-id1517256\">where<\/p>\n<span id=\"eip-id1165135512554\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154155\/9.8.2.jpg\" alt=\"\"><\/span>\n<p id=\"fs-id1702973\">If we are writing the determinant[latex]\\,{D}_{x},[\/latex]we replace the[latex]\\,x\\,[\/latex]column with the constant column. If we are writing the determinant[latex]{D}_{y},[\/latex]we replace the[latex]\\,y\\,[\/latex]column with the constant column. If we are writing the determinant[latex]\\,{D}_{z},[\/latex]we replace the[latex]\\,z\\,[\/latex]column with the constant column. Always check the answer.<\/p>\n\n<div id=\"Example_09_08_04\" class=\"textbox examples\">\n<div id=\"fs-id1563720\">\n<div id=\"fs-id1182614\">\n<h3>Solving a 3 \u00d7 3 System Using Cramer\u2019s Rule<\/h3>\n<p id=\"fs-id1182620\">Find the solution to the given 3 \u00d7 3 system using Cramer\u2019s Rule.<\/p>\n\n<div id=\"fs-id1182624\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}x+y-z=6\\\\ 3x-2y+z=-5\\\\ x+3y-2z=14\\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"1588465\"]Show Solution[\/reveal-answer][hidden-answer a=\"1588465\"]\n\nUse Cramer\u2019s Rule.\n<div id=\"fs-id1406667\" class=\"unnumbered aligncenter\">[latex]D=|\\begin{array}{ccc}1&amp; \\,\\,1&amp; -1\\\\ 3&amp; -2&amp; \\,\\,\\,1\\\\ 1&amp; \\,\\,3&amp; -2\\end{array}|,{D}_{x}=|\\begin{array}{ccc}6&amp; 1&amp; -1\\\\ -5&amp; -2&amp; \\,\\,\\,1\\\\ 14&amp; \\,\\,3&amp; -2\\end{array}|,{D}_{y}=|\\begin{array}{ccc}1&amp; \\,6&amp; -1\\\\ 3&amp; -5&amp; \\,\\,1\\\\ 1&amp; 14&amp; -2\\end{array}|,{D}_{z}=|\\begin{array}{ccc}1&amp; \\,1&amp; 6\\\\ 3&amp; -2&amp; -5\\\\ 1&amp; \\,\\,3&amp; 14\\end{array}|[\/latex]<\/div>\n<p id=\"fs-id1436106\">Then,<\/p>\n\n<div id=\"fs-id1436109\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}x=\\frac{{D}_{x}}{D}=\\frac{-3}{-3}=1\\hfill \\\\ y=\\frac{{D}_{y}}{D}=\\frac{-9}{-3}=3\\hfill \\\\ z=\\frac{{D}_{z}}{D}=\\frac{6}{-3}=-2\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1936367\">The solution is[latex]\\left(1,3,-2\\right).[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1460684\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_09_08_03\">\n<div id=\"fs-id1446687\">\n<p id=\"fs-id1446688\">Use Cramer\u2019s Rule to solve the 3 \u00d7 3 matrix.<\/p>\n\n<div id=\"fs-id1446691\">[latex]\\begin{array}{r}\\hfill x-3y+7z=13\\\\ \\hfill x+y+z=1\\,\\,\\,\\\\ \\hfill x-2y+3z=4\\,\\,\\,\\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1405450\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1405450\"]\n<p id=\"fs-id1405450\">[latex]\\left(-2,\\frac{3}{5},\\frac{12}{5}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_09_08_05\" class=\"textbox examples\">\n<div id=\"fs-id1698619\">\n<div id=\"fs-id1698622\">\n<h3>Using Cramer\u2019s Rule to Solve an Inconsistent System<\/h3>\n<p id=\"fs-id1658471\">Solve the system of equations using Cramer\u2019s Rule.<\/p>\n\n<div id=\"fs-id1658475\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}3x-2y=4\\text{\u2003}\\left(1\\right)\\\\ 6x-4y=0\\text{\u2003}\\left(2\\right)\\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1464455\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1464455\"]\n<p id=\"fs-id1464455\">We begin by finding the determinants[latex]\\,D,{D}_{x},\\text{and }{D}_{y}.[\/latex]<\/p>\n\n<div id=\"fs-id1440950\" class=\"unnumbered aligncenter\">[latex]D=|\\begin{array}{cc}3&amp; -2\\\\ 6&amp; -4\\end{array}|=3\\left(-4\\right)-6\\left(-2\\right)=0[\/latex]<\/div>\n<p id=\"fs-id2651884\">We know that a determinant of zero means that either the system has no solution or it has an infinite number of solutions. To see which one, we use the process of elimination. Our goal is to eliminate one of the variables.<\/p>\n\n<ol id=\"fs-id2651889\" type=\"1\">\n \t<li>Multiply equation (1) by[latex]\\,-2.[\/latex]<\/li>\n \t<li>Add the result to equation[latex]\\,\\left(2\\right).[\/latex]<\/li>\n<\/ol>\n<div id=\"fs-id1351077\">[latex]\\begin{array}{l}\\underset{\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_}{\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ -6x+4y\\,\\,\\,\\,=-8\\hfill \\end{array}\\hfill \\\\ \\,\\,\\,6x-4y\\,\\,\\,\\,\\,\\,=\\,\\,\\,\\,0\\hfill \\end{array}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,0\\,\\,\\,\\,\\,\\,=\\,-8\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1352000\">We obtain the equation[latex]\\,0=-8,\\,[\/latex]which is false. Therefore, the system has no solution. Graphing the system reveals two parallel lines. See <a class=\"autogenerated-content\" href=\"#Figure_09_08_003\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_09_08_003\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154202\/CNX_Precalc_Figure_09_08_003.jpg\" alt=\"\" width=\"487\" height=\"441\"> <strong>Figure 1.<\/strong>[\/caption]\n\n<\/div>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_09_08_06\" class=\"textbox examples\">\n<div id=\"fs-id1694152\">\n<div id=\"fs-id1694154\">\n<h3>Use Cramer\u2019s Rule to Solve a Dependent System<\/h3>\n<p id=\"fs-id1694160\">Solve the system with an infinite number of solutions.<\/p>\n\n<div id=\"fs-id1694163\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{rr}\\hfill x-2y+3z=0&amp; \\hfill \\left(1\\right)\\\\ \\hfill 3x+y-2z=0&amp; \\hfill \\left(2\\right)\\\\ \\hfill 2x-4y+6z=0&amp; \\hfill \\left(3\\right)\\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1360831\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1360831\"]\n<p id=\"fs-id1360834\">Let\u2019s find the determinant first. Set up a matrix augmented by the first two columns.<\/p>\n\n<div id=\"fs-id1360839\" class=\"unnumbered aligncenter\">[latex]|\\begin{array}{rrr}\\hfill 1&amp; \\hfill -2&amp; \\hfill 3\\\\ \\hfill 3&amp; \\hfill 1&amp; \\hfill -2\\\\ \\hfill 2&amp; \\hfill -4&amp; \\hfill 6\\end{array}\\text{ }|\\text{ }\\begin{array}{rr}\\hfill 1&amp; \\hfill -2\\\\ \\hfill 3&amp; \\hfill 1\\\\ \\hfill 2&amp; \\hfill -4\\end{array}|[\/latex]<\/div>\n<p id=\"fs-id1523859\">Then,<\/p>\n\n<div id=\"fs-id1511748\" class=\"unnumbered aligncenter\">[latex]1\\left(1\\right)\\left(6\\right)+\\left(-2\\right)\\left(-2\\right)\\left(2\\right)+3\\left(3\\right)\\left(-4\\right)-2\\left(1\\right)\\left(3\\right)-\\left(-4\\right)\\left(-2\\right)\\left(1\\right)-6\\left(3\\right)\\left(-2\\right)=0[\/latex]<\/div>\n<p id=\"fs-id1699565\">As the determinant equals zero, there is either no solution or an infinite number of solutions. We have to perform elimination to find out.<\/p>\n\n<ol id=\"fs-id1456188\" type=\"1\">\n \t<li>Multiply equation (1) by[latex]\\,-2\\,[\/latex]and add the result to equation (3):\n<div id=\"eip-id1165132152839\" class=\"unnumbered\">[latex]\\frac{\\begin{array}{r}\\hfill -2x+4y-6x=0\\\\ \\hfill 2x-4y+6z=0\\end{array}}{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,0=0}[\/latex]<\/div><\/li>\n \t<li>Obtaining an answer of[latex]\\,0=0,\\,[\/latex]a statement that is always true, means that the system has an infinite number of solutions. Graphing the system, we can see that two of the planes are the same and they both intersect the third plane on a line. See <a class=\"autogenerated-content\" href=\"#Figure_09_08_005\">(Figure).<\/a>\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154205\/CNX_Precalc_Figure_09_08_005.jpg\" alt=\"\" width=\"487\" height=\"214\"> <strong>Figure 2.<\/strong>[\/caption]\n\n[\/hidden-answer]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1651973\" class=\"bc-section section\">\n<h3>Understanding Properties of Determinants<\/h3>\n<p id=\"fs-id1651978\">There are many <span class=\"no-emphasis\">properties of determinants<\/span>. Listed here are some properties that may be helpful in calculating the determinant of a matrix.<\/p>\n\n<div id=\"fs-id1651986\">\n<h3>Properties of Determinants<\/h3>\n<ol id=\"fs-id1459457\" type=\"1\">\n \t<li>If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.<\/li>\n \t<li>When two rows are interchanged, the determinant changes sign.<\/li>\n \t<li>If either two rows or two columns are identical, the determinant equals zero.<\/li>\n \t<li>If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.<\/li>\n \t<li>The determinant of an inverse matrix[latex]\\,{A}^{-1}\\,[\/latex]is the reciprocal of the determinant of the matrix[latex]\\,A.[\/latex]<\/li>\n \t<li>If any row or column is multiplied by a constant, the determinant is multiplied by the same factor.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_09_08_07\" class=\"textbox examples\">\n<div id=\"fs-id1531866\">\n<div id=\"fs-id1531868\">\n<h3>Illustrating Properties of Determinants<\/h3>\n<p id=\"fs-id1531874\">Illustrate each of the properties of determinants.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1531880\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1531880\"]\n<p id=\"fs-id1531880\">Property 1 states that if the matrix is in upper triangular form, the determinant is the product of the entries down the main diagonal.<\/p>\n\n<div id=\"fs-id1685740\" class=\"unnumbered aligncenter\">[latex]A=\\left[\\begin{array}{rrr}\\hfill 1&amp; \\hfill \\,\\,2&amp; \\hfill 3\\\\ \\hfill 0&amp; \\hfill \\,\\,2&amp; \\hfill 1\\\\ \\hfill 0&amp; \\hfill \\,\\,0&amp; \\hfill -1\\end{array}\\right][\/latex]<\/div>\n<p id=\"fs-id1370716\">Augment[latex]\\,A\\,[\/latex]with the first two columns.<\/p>\n\n<div id=\"fs-id1504903\" class=\"unnumbered aligncenter\">[latex]A=\\left[\\begin{array}{ccc}1&amp; 2&amp; 3\\\\ 0&amp; 2&amp; 1\\\\ 0&amp; 0&amp; -1\\end{array}|\\,\\,\\,\\begin{array}{c}1\\\\ 0\\\\ 0\\end{array}\\,\\,\\,\\,\\begin{array}{c}2\\\\ 2\\\\ 0\\end{array}\\right][\/latex]<\/div>\n<p id=\"fs-id1608365\">Then<\/p>\n\n<div id=\"fs-id1608368\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\mathrm{det}\\left(A\\right)=1\\left(2\\right)\\left(-1\\right)+2\\left(1\\right)\\left(0\\right)+3\\left(0\\right)\\left(0\\right)-0\\left(2\\right)\\left(3\\right)-0\\left(1\\right)\\left(1\\right)+1\\left(0\\right)\\left(2\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=-2\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1429354\">Property 2 states that interchanging rows changes the sign. Given<\/p>\n\n<div class=\"unnumbered\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ A=\\left[\\begin{array}{cc}-1&amp; 5\\\\ 4&amp; -3\\end{array}\\right],\\,\\,\\mathrm{det}\\left(A\\right)=\\left(-1\\right)\\left(-3\\right)-\\left(4\\right)\\left(5\\right)=3-20=-17\\end{array}\\hfill \\\\ \\hfill \\\\ B=\\left[\\begin{array}{cc}4&amp; -3\\\\ -1&amp; 5\\end{array}\\right],\\,\\,\\mathrm{det}\\left(B\\right)=\\left(4\\right)\\left(5\\right)-\\left(-1\\right)\\left(-3\\right)=20-3=17\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1433736\">Property 3 states that if two rows or two columns are identical, the determinant equals zero.<\/p>\n\n<div class=\"unnumbered\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,A=\\left[\\begin{array}{ccc}1&amp; 2&amp; 2\\\\ 2&amp; 2&amp; 2\\\\ -1&amp; 2&amp; 2\\end{array}\\text{ }|\\text{ }\\begin{array}{c}1\\\\ 2\\\\ -1\\end{array} \\begin{array}{c}2\\\\ 2\\\\ 2\\end{array}\\right]\\hfill \\\\ \\hfill \\\\ \\mathrm{det}\\left(A\\right)=1\\left(2\\right)\\left(2\\right)+2\\left(2\\right)\\left(-1\\right)+2\\left(2\\right)\\left(2\\right)+1\\left(2\\right)\\left(2\\right)-2\\left(2\\right)\\left(1\\right)-2\\left(2\\right)\\left(2\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=4-4+8+4-4-8=0\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1374562\">Property 4 states that if a row or column equals zero, the determinant equals zero. Thus,<\/p>\n\n<div id=\"fs-id1408472\" class=\"unnumbered aligncenter\">[latex]A=\\left[\\begin{array}{cc}1&amp; 2\\\\ 0&amp; 0\\end{array}\\right],\\,\\,\\,\\mathrm{det}\\left(A\\right)=1\\left(0\\right)-2\\left(0\\right)=0[\/latex]<\/div>\n<p id=\"fs-id1434974\">Property 5 states that the determinant of an inverse matrix[latex]\\,{A}^{-1}\\,[\/latex]is the reciprocal of the determinant[latex]\\,A.\\,[\/latex]Thus,<\/p>\n\n<div id=\"fs-id1371025\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,A=\\left[\\begin{array}{cc}1&amp; 2\\\\ 3&amp; 4\\end{array}\\right],\\mathrm{det}\\left(A\\right)=1\\left(4\\right)-3\\left(2\\right)=-2\\hfill \\\\ \\hfill \\\\ {A}^{-1}=\\left[\\begin{array}{cc}-2&amp; 1\\\\ \\frac{3}{2}&amp; -\\frac{1}{2}\\end{array}\\right],\\mathrm{det}\\left({A}^{-1}\\right)=-2\\left(-\\frac{1}{2}\\right)-\\left(\\frac{3}{2}\\right)\\left(1\\right)=-\\frac{1}{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1455730\">Property 6 states that if any row or column of a matrix is multiplied by a constant, the determinant is multiplied by the same factor. Thus,<\/p>\n\n<div id=\"fs-id1455735\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}A=\\left[\\begin{array}{cc}1&amp; 2\\\\ 3&amp; 4\\end{array}\\right],\\mathrm{det}\\left(A\\right)=1\\left(4\\right)-2\\left(3\\right)=-2\\hfill \\\\ \\hfill \\\\ B=\\left[\\begin{array}{cc}2\\left(1\\right)&amp; 2\\left(2\\right)\\\\ 3&amp; 4\\end{array}\\right],\\mathrm{det}\\left(B\\right)=2\\left(4\\right)-3\\left(4\\right)=-4\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_09_08_08\" class=\"textbox examples\">\n<div id=\"fs-id1417083\">\n<div id=\"fs-id1417085\">\n<h3>Using Cramer\u2019s Rule and Determinant Properties to Solve a System<\/h3>\n<p id=\"fs-id1417091\">Find the solution to the given 3 \u00d7 3 system.<\/p>\n\n<div id=\"fs-id1417094\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}2x+4y+4z=2\\hfill &amp; \\left(1\\right)\\hfill \\\\ 3x+7y+7z=-5\\hfill &amp; \\left(2\\right)\\hfill \\\\ \\text{ }x+2y+2z=4\\hfill &amp; \\left(3\\right)\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1653554\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1653554\"]\n<p id=\"fs-id1653554\">Using <span class=\"no-emphasis\">Cramer\u2019s Rule<\/span>, we have<\/p>\n\n<div id=\"fs-id1407186\" class=\"unnumbered aligncenter\">[latex]D=|\\begin{array}{ccc}2&amp; 4&amp; 4\\\\ 3&amp; 7&amp; 7\\\\ 1&amp; 2&amp; 2\\end{array}|[\/latex]<\/div>\n<p id=\"fs-id1676090\">Notice that the second and third columns are identical. According to Property 3, the determinant will be zero, so there is either no solution or an infinite number of solutions. We have to perform elimination to find out.<\/p>\n\n<ol id=\"fs-id1676095\" type=\"1\">\n \t<li>Multiply equation (3) by \u20132 and add the result to equation (1).\n<div id=\"fs-id1407844\" class=\"unnumbered aligncenter\">[latex]\\frac{\\begin{array}{l}-2x-4y-4x=-8\\hfill \\\\ \\text{ }2x+4y+4z=2\\,\\,\\,\\,\\,\\hfill \\end{array}}{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,0=-6}[\/latex]<\/div><\/li>\n<\/ol>\n<p id=\"fs-id1422312\">Obtaining a statement that is a contradiction means that the system has no solution.[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1422319\" class=\"precalculus media\">\n<p id=\"fs-id1677169\">Access these online resources for additional instruction and practice with Cramer\u2019s Rule.<\/p>\n\n<ul id=\"fs-id1677174\">\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/system2cramer\">Solve a System of Two Equations Using Cramer's Rule<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/system3cramer\">Solve a Systems of Three Equations using Cramer's Rule<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1677186\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1600273\">\n \t<li>The determinant for[latex]\\,\\left[\\begin{array}{cc}a&amp; b\\\\ c&amp; d\\end{array}\\right]\\,[\/latex]is[latex]\\,ad-bc.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_09_08_01\">(Figure)<\/a>.<\/li>\n \t<li>Cramer\u2019s Rule replaces a variable column with the constant column. Solutions are[latex]\\,x=\\frac{{D}_{x}}{D},y=\\frac{{D}_{y}}{D}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_09_08_02\">(Figure)<\/a>.<\/li>\n \t<li>To find the determinant of a 3\u00d73 matrix, augment with the first two columns. Add the three diagonal entries (upper left to lower right) and subtract the three diagonal entries (lower left to upper right). See <a class=\"autogenerated-content\" href=\"#Example_09_08_03\">(Figure)<\/a>.<\/li>\n \t<li>To solve a system of three equations in three variables using Cramer\u2019s Rule, replace a variable column with the constant column for each desired solution:[latex]\\,x=\\frac{{D}_{x}}{D},y=\\frac{{D}_{y}}{D},z=\\frac{{D}_{z}}{D}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_09_08_04\">(Figure)<\/a>.<\/li>\n \t<li>Cramer\u2019s Rule is also useful for finding the solution of a system of equations with no solution or infinite solutions. See <a class=\"autogenerated-content\" href=\"#Example_09_08_05\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_09_08_06\">(Figure)<\/a>.<\/li>\n \t<li>Certain properties of determinants are useful for solving problems. For example:\n<ul id=\"eip-id1165135344724\">\n \t<li>If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.<\/li>\n \t<li>When two rows are interchanged, the determinant changes sign.<\/li>\n \t<li>If either two rows or two columns are identical, the determinant equals zero.<\/li>\n \t<li>If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.<\/li>\n \t<li>The determinant of an inverse matrix[latex]\\,{A}^{-1}\\,[\/latex]is the reciprocal of the determinant of the matrix[latex]\\,A.[\/latex]<\/li>\n \t<li>If any row or column is multiplied by a constant, the determinant is multiplied by the same factor. See <a class=\"autogenerated-content\" href=\"#Example_09_08_07\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_09_08_08\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1408613\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1408620\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1405255\">\n<div id=\"fs-id1405256\">\n<p id=\"fs-id1405257\">Explain why we can always evaluate the determinant of a square matrix.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"1405260\"]Show Solution[\/reveal-answer][hidden-answer a=\"1405260\"]\n\nA determinant is the sum and products of the entries in the matrix, so you can always evaluate that product\u2014even if it does end up being 0.\n\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1405269\">\n<div id=\"fs-id1405270\">\n<p id=\"fs-id1405271\">Examining Cramer\u2019s Rule, explain why there is no unique solution to the system when the determinant of your matrix is 0. For simplicity, use a[latex]\\,2\\,\u00d7\\,2\\,[\/latex]matrix.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1394594\">\n<div id=\"fs-id1394595\">\n<p id=\"fs-id1394596\">Explain what it means in terms of an inverse for a matrix to have a 0 determinant.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1394601\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1394601\"]\n<p id=\"fs-id1394601\">The inverse does not exist.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1394606\">\n<div id=\"fs-id1394607\">\n<p id=\"fs-id1394608\">The determinant of[latex]\\,2\\,\u00d7\\,2\\,[\/latex]matrix[latex]\\,A\\,[\/latex]is 3. If you switch the rows and multiply the first row by 6 and the second row by 2, explain how to find the determinant and provide the answer.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1417224\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1417229\">For the following exercises, find the determinant.<\/p>\n\n<div id=\"fs-id1417233\">\n<div id=\"fs-id1417234\">\n<p id=\"fs-id1394308\">[latex]|\\begin{array}{cc}1&amp; 2\\\\ 3&amp; 4\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1394170\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1394170\"]\n<p id=\"fs-id1394170\">[latex]-2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1522034\">\n<div id=\"fs-id1522035\">\n<p id=\"fs-id1522036\">[latex]|\\begin{array}{rr}\\hfill -1&amp; \\hfill 2\\\\ \\hfill 3&amp; \\hfill -4\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1434436\">\n<div id=\"fs-id1434437\">\n<p id=\"fs-id1434438\">[latex]|\\begin{array}{rr}\\hfill 2&amp; \\hfill -5\\\\ \\hfill -1&amp; \\hfill 6\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1430900\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1430900\"]\n<p id=\"fs-id1430900\">[latex]7[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1430911\">\n<div id=\"fs-id1430912\">\n<p id=\"fs-id1430913\">[latex]|\\begin{array}{cc}-8&amp; 4\\\\ -1&amp; 5\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1677911\">\n<div id=\"fs-id1677912\">\n<p id=\"fs-id1677913\">[latex]|\\begin{array}{rr}\\hfill 1&amp; \\hfill 0\\\\ \\hfill 3&amp; \\hfill -4\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1395220\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1395220\"]\n<p id=\"fs-id1395220\">[latex]-4[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1408793\">\n<div id=\"fs-id1408794\">\n<p id=\"fs-id1408795\">[latex]|\\begin{array}{rr}\\hfill 10&amp; \\hfill 20\\\\ \\hfill 0&amp; \\hfill -10\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1407150\">\n<div id=\"fs-id1407151\">\n<p id=\"fs-id1407152\">[latex]|\\begin{array}{cc}10&amp; 0.2\\\\ 5&amp; 0.1\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1660075\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1660075\"]\n<p id=\"fs-id1660075\">[latex]0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1660084\">\n<div id=\"fs-id1660085\">\n<p id=\"fs-id1660086\">[latex]|\\begin{array}{rr}\\hfill 6&amp; \\hfill -3\\\\ \\hfill 8&amp; \\hfill 4\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1587739\">\n<div id=\"fs-id1587740\">\n<p id=\"fs-id1587741\">[latex]|\\begin{array}{rr}\\hfill -2&amp; \\hfill -3\\\\ \\hfill 3.1&amp; \\hfill 4,000\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id885813\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id885813\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id885813\"]\n<p id=\"fs-id885815\">[latex]-7,990.7[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div>[latex]|\\begin{array}{rr}\\hfill -1.1&amp; \\hfill 0.6\\\\ \\hfill 7.2&amp; \\hfill -0.5\\end{array}|[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1513986\">\n<div id=\"fs-id1513987\">\n<p id=\"fs-id1513988\">[latex]|\\begin{array}{rrr}\\hfill -1&amp; \\hfill 0&amp; \\hfill 0\\\\ \\hfill 0&amp; \\hfill 1&amp; \\hfill 0\\\\ \\hfill 0&amp; \\hfill 0&amp; \\hfill -3\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1339464\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1339464\"]\n<p id=\"fs-id1339464\">[latex]3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1339474\">\n<div id=\"fs-id1339475\">\n<p id=\"fs-id1339476\">[latex]|\\begin{array}{rrr}\\hfill -1&amp; \\hfill 4&amp; \\hfill 0\\\\ \\hfill 0&amp; \\hfill 2&amp; \\hfill 3\\\\ \\hfill 0&amp; \\hfill 0&amp; \\hfill -3\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1586650\">\n<div id=\"fs-id1586652\">\n<p id=\"fs-id1586653\">[latex]|\\begin{array}{ccc}1&amp; 0&amp; 1\\\\ 0&amp; 1&amp; 0\\\\ 1&amp; 0&amp; 0\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1601846\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1601846\"]\n<p id=\"fs-id1601846\">[latex]-1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id881216\">\n<div id=\"fs-id881217\">\n<p id=\"fs-id881218\">[latex]|\\begin{array}{rrr}\\hfill 2&amp; \\hfill -3&amp; \\hfill 1\\\\ \\hfill 3&amp; \\hfill -4&amp; \\hfill 1\\\\ \\hfill -5&amp; \\hfill 6&amp; \\hfill 1\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2692384\">\n<div id=\"fs-id2692386\">\n<p id=\"fs-id2692387\">[latex]|\\begin{array}{rrr}\\hfill -2&amp; \\hfill 1&amp; \\hfill 4\\\\ \\hfill -4&amp; \\hfill 2&amp; \\hfill -8\\\\ \\hfill 2&amp; \\hfill -8&amp; \\hfill -3\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1663783\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1663783\"]\n<p id=\"fs-id1663783\">[latex]224[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1663796\">\n<div id=\"fs-id1663797\">\n<p id=\"fs-id1663798\">[latex]|\\begin{array}{rrr}\\hfill 6&amp; \\hfill -1&amp; \\hfill 2\\\\ \\hfill -4&amp; \\hfill -3&amp; \\hfill 5\\\\ \\hfill 1&amp; \\hfill 9&amp; \\hfill -1\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1365782\">\n<div id=\"fs-id1365783\">\n<p id=\"fs-id1365784\">[latex]|\\begin{array}{rrr}\\hfill 5&amp; \\hfill 1&amp; \\hfill -1\\\\ \\hfill 2&amp; \\hfill 3&amp; \\hfill 1\\\\ \\hfill 3&amp; \\hfill -6&amp; \\hfill -3\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1616218\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1616218\"]\n<p id=\"fs-id1616218\">[latex]15[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1616230\">\n<div id=\"fs-id1616232\">\n<p id=\"fs-id1616233\">[latex]|\\begin{array}{rrr}\\hfill 1.1&amp; \\hfill 2&amp; \\hfill -1\\\\ \\hfill -4&amp; \\hfill 0&amp; \\hfill 0\\\\ \\hfill 4.1&amp; \\hfill -0.4&amp; \\hfill 2.5\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1370226\">\n<div id=\"fs-id1370227\">\n<p id=\"fs-id1370228\">[latex]|\\begin{array}{rrr}\\hfill 2&amp; \\hfill -1.6&amp; \\hfill 3.1\\\\ \\hfill 1.1&amp; \\hfill 3&amp; \\hfill -8\\\\ \\hfill -9.3&amp; \\hfill 0&amp; \\hfill 2\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1534877\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1534877\"]\n<p id=\"fs-id1534877\">[latex]-17.03[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1534892\">\n<div id=\"fs-id1534893\">\n<p id=\"fs-id1534894\">[latex]|\\begin{array}{ccc}-\\frac{1}{2}&amp; \\frac{1}{3}&amp; \\frac{1}{4}\\\\ \\frac{1}{5}&amp; -\\frac{1}{6}&amp; \\frac{1}{7}\\\\ 0&amp; 0&amp; \\frac{1}{8}\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1891694\">For the following exercises, solve the system of linear equations using Cramer\u2019s Rule.<\/p>\n\n<div id=\"fs-id1891699\">\n<div id=\"fs-id1891700\">\n<p id=\"fs-id1891701\">[latex]\\begin{array}{l}2x-3y=-1\\\\ 4x+5y=9\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1694937\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1694937\"]\n<p id=\"fs-id1694937\">[latex]\\left(1,1\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1914122\">\n<div id=\"fs-id1914123\">\n<p id=\"fs-id1914124\">[latex]\\begin{array}{r}5x-4y=2\\\\ -4x+7y=6\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1704920\">\n<div id=\"fs-id1704921\">\n<p id=\"fs-id1704922\">[latex]\\begin{array}{l}\\text{ }6x-3y=2\\,\\,\\,\\,\\,\\hfill \\\\ -8x+9y=-1\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1433846\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1433846\"]\n<p id=\"fs-id1433846\">[latex]\\left(\\frac{1}{2},\\frac{1}{3}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1381750\">\n<div id=\"fs-id1675292\">\n<p id=\"fs-id1675294\">[latex]\\begin{array}{l}2x+6y=12\\\\ 5x-2y=13\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1408543\">\n<div id=\"fs-id1408544\">\n<p id=\"fs-id1408546\">[latex]\\begin{array}{l}4x+3y=23\\,\\,\\hfill \\\\ \\text{ }2x-y=-1\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1516926\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1516926\"]\n<p id=\"fs-id1516926\">[latex]\\left(2,5\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1513467\">\n<div id=\"fs-id1513468\">\n<p id=\"fs-id1513469\">[latex]\\begin{array}{l}10x-6y=2\\,\\,\\,\\,\\hfill \\\\ -5x+8y=-1\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1512009\">\n<div id=\"fs-id1512010\">\n<p id=\"fs-id1512012\">[latex]\\begin{array}{l}4x-3y=-3\\\\ 2x+6y=-4\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1615360\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1615360\"]\n<p id=\"fs-id1615360\">[latex]\\left(-1,-\\frac{1}{3}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1523200\">\n<div id=\"fs-id1514396\">\n<p id=\"fs-id1514397\">[latex]\\begin{array}{r}4x-5y=7\\\\ -3x+9y=0\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1701646\">\n<div id=\"fs-id1701647\">\n<p id=\"fs-id1701648\">[latex]\\begin{array}{l}4x+10y=180\\,\\,\\,\\,\\hfill \\\\ -3x-5y=-105\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1530071\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1530071\"]\n<p id=\"fs-id1530071\">[latex]\\left(15,12\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1394699\">\n<div id=\"fs-id1394700\">\n<p id=\"fs-id1394702\">[latex]\\begin{array}{l}\\text{ }8x-2y=-3\\hfill \\\\ -4x+6y=4\\,\\,\\,\\,\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1517129\">For the following exercises, solve the system of linear equations using Cramer\u2019s Rule.<\/p>\n\n<div id=\"fs-id1517134\">\n<div id=\"fs-id1517135\">[latex]\\begin{array}{l}\\text{ }x+2y-4z=-1\\hfill \\\\ \\text{ }7x+3y+5z=26\\,\\,\\hfill \\\\ -2x-6y+7z=-6\\hfill \\end{array}[\/latex]<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1530969\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1530969\"]\n<p id=\"fs-id1530969\">[latex]\\left(1,3,2\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1699396\">\n<div id=\"fs-id1699397\">\n<p id=\"fs-id1699398\">[latex]\\begin{array}{l}-5x+2y-4z=-47\\hfill \\\\ \\text{ }4x-3y-z=-94\\hfill \\\\ \\text{ }3x-3y+2z=94\\,\\,\\,\\,\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1351513\">\n<div id=\"fs-id1351514\">\n<p id=\"fs-id1351515\">[latex]\\begin{array}{l}\\text{ }4x+5y-z=-7\\hfill \\\\ -2x-9y+2z=8\\,\\,\\,\\,\\hfill \\\\ \\text{ }5y+7z=21\\,\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1699357\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1699357\"]\n<p id=\"fs-id1699357\">[latex]\\left(-1,0,3\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1355045\">\n<div id=\"fs-id1355046\">\n<p id=\"fs-id1355047\">[latex]\\begin{array}{r}4x-3y+4z=10\\\\ 5x-2z=-2\\\\ 3x+2y-5z=-9\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1410018\">\n<div id=\"fs-id1410019\">\n<p id=\"fs-id1410020\">[latex]\\begin{array}{l}4x-2y+3z=6\\,\\,\\,\\hfill \\\\ \\text{ }-6x+y=-2\\hfill \\\\ 2x+7y+8z=24\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1506671\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1506671\"]\n<p id=\"fs-id1506671\">[latex]\\left(\\frac{1}{2},1,2\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1430552\">\n<div id=\"fs-id1430553\">\n<p id=\"fs-id1430554\">[latex]\\begin{array}{r}\\hfill 5x+2y-z=1\\,\\,\\,\\,\\,\\\\ \\hfill -7x-8y+3z=1.5\\\\ \\hfill 6x-12y+z=7\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1348575\">\n<div id=\"fs-id1348576\">\n<p id=\"fs-id1348577\">[latex]\\begin{array}{l}\\text{ }13x-17y+16z=73\\,\\,\\,\\,\\hfill \\\\ -11x+15y+17z=61\\,\\,\\,\\,\\hfill \\\\ \\text{ }46x+10y-30z=-18\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"1433664\"]Show Solution[\/reveal-answer][hidden-answer a=\"1433664\"]\n[latex]\\left(2,1,4\\right)[\/latex][\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1519001\">\n<div id=\"fs-id1519002\">\n<p id=\"fs-id1519003\">[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ -4x-3y-8z=-7\\hfill \\end{array}\\hfill \\\\ \\text{ }2x-9y+5z=0.5\\hfill \\\\ \\text{ }5x-6y-5z=-2\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1601296\">\n<div id=\"fs-id1601297\">\n<p id=\"fs-id1601298\">[latex]\\begin{array}{l}\\text{ }4x-6y+8z=10\\,\\,\\hfill \\\\ -2x+3y-4z=-5\\hfill \\\\ \\text{ }x+y+z=1\\,\\,\\,\\,\\,\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1673950\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1673950\"]\n<p id=\"fs-id1673950\">Infinite solutions<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1446784\">\n<div id=\"fs-id1446785\">\n<p id=\"fs-id1446786\">[latex]\\begin{array}{r}\\hfill 4x-6y+8z=10\\,\\,\\,\\,\\,\\\\ \\hfill -2x+3y-4z=-5\\,\\,\\,\\\\ \\hfill 12x+18y-24z=-30\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1487525\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1487530\">For the following exercises, use the determinant function on a graphing utility.<\/p>\n\n<div id=\"fs-id1487533\">\n<div id=\"fs-id1487534\">[latex]|\\begin{array}{rrrr}\\hfill 1&amp; \\hfill 0&amp; \\hfill 8&amp; \\hfill 9\\\\ \\hfill 0&amp; \\hfill 2&amp; \\hfill 1&amp; \\hfill 0\\\\ \\hfill 1&amp; \\hfill 0&amp; \\hfill 3&amp; \\hfill 0\\\\ \\hfill 0&amp; \\hfill 2&amp; \\hfill 4&amp; \\hfill 3\\end{array}|[\/latex]<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1597711\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1597711\"]\n<p id=\"fs-id1597711\">[latex]24[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1597725\">\n<div id=\"fs-id1597726\">\n<p id=\"fs-id1597727\">[latex]|\\begin{array}{rrrr}\\hfill 1&amp; \\hfill 0&amp; \\hfill 2&amp; \\hfill 1\\\\ \\hfill 0&amp; \\hfill -9&amp; \\hfill 1&amp; \\hfill 3\\\\ \\hfill 3&amp; \\hfill 0&amp; \\hfill -2&amp; \\hfill -1\\\\ \\hfill 0&amp; \\hfill 1&amp; \\hfill 1&amp; \\hfill -2\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1700780\">\n<div id=\"fs-id1700781\">\n<p id=\"fs-id1700782\">[latex]|\\begin{array}{rrrr}\\hfill \\frac{1}{2}&amp; \\hfill 1&amp; \\hfill 7&amp; \\hfill 4\\\\ \\hfill 0&amp; \\hfill \\frac{1}{2}&amp; \\hfill 100&amp; \\hfill 5\\\\ \\hfill 0&amp; \\hfill 0&amp; \\hfill 2&amp; \\hfill 2,000\\\\ \\hfill 0&amp; \\hfill 0&amp; \\hfill 0&amp; \\hfill 2\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1905255\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1905255\"]\n<p id=\"fs-id1905255\">[latex]1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1905264\">\n<div id=\"fs-id1905265\">\n<p id=\"fs-id1905266\">[latex]|\\begin{array}{rrrr}\\hfill 1&amp; \\hfill 0&amp; \\hfill 0&amp; \\hfill 0\\\\ \\hfill 2&amp; \\hfill 3&amp; \\hfill 0&amp; \\hfill 0\\\\ \\hfill 4&amp; \\hfill 5&amp; \\hfill 6&amp; \\hfill 0\\\\ \\hfill 7&amp; \\hfill 8&amp; \\hfill 9&amp; \\hfill 0\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1420602\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<p id=\"fs-id1702572\">For the following exercises, create a system of linear equations to describe the behavior. Then, calculate the determinant. Will there be a unique solution? If so, find the unique solution.<\/p>\n\n<div id=\"fs-id1702577\">\n<div id=\"fs-id1702578\">\n<p id=\"fs-id1702579\">Two numbers add up to 56. One number is 20 less than the other.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1702584\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1702584\"]\n<p id=\"fs-id1702584\">Yes; 18, 38<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1702588\">\n<div id=\"fs-id1702590\">\n<p id=\"fs-id1702591\">Two numbers add up to 104. If you add two times the first number plus two times the second number, your total is 208<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1702595\">\n<div id=\"fs-id1791156\">\n<p id=\"fs-id1791157\">Three numbers add up to 106. The first number is 3 less than the second number. The third number is 4 more than the first number.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1791164\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1791164\"]\n<p id=\"fs-id1791164\">Yes; 33, 36, 37<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1791168\">\n<div id=\"fs-id1791169\">\n<p id=\"fs-id1791170\">Three numbers add to 216. The sum of the first two numbers is 112. The third number is 8 less than the first two numbers combined.<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1791174\">For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer\u2019s Rule.<\/p>\n\n<div id=\"fs-id1791180\">\n<div id=\"fs-id1791181\">\n<p id=\"fs-id1791182\">You invest $10,000 into two accounts, which receive 8% interest and 5% interest. At the end of a year, you had $10,710 in your combined accounts. How much was invested in each account?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1354932\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1354932\"]\n<p id=\"fs-id1354932\">$7,000 in first account, $3,000 in second account.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1354936\">\n<div id=\"fs-id1354937\">\n<p id=\"fs-id1354938\">You invest $80,000 into two accounts, $22,000 in one account, and $58,000 in the other account. At the end of one year, assuming simple interest, you have earned $2,470 in interest. The second account receives half a percent less than twice the interest on the first account. What are the interest rates for your accounts?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1354944\">\n<div id=\"fs-id1354946\">\n<p id=\"fs-id1354947\">A movie theater needs to know how many adult tickets and children tickets were sold out of the 1,200 total tickets. If children\u2019s tickets are $5.95, adult tickets are $11.15, and the total amount of revenue was $12,756, how many children\u2019s tickets and adult tickets were sold?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1584198\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1584198\"]\n<p id=\"fs-id1584198\">120 children, 1,080 adult<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1584202\">\n<div id=\"fs-id1584203\">\n<p id=\"fs-id1584204\">A concert venue sells single tickets for $40 each and couple\u2019s tickets for $65. If the total revenue was $18,090 and the 321 tickets were sold, how many single tickets and how many couple\u2019s tickets were sold?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1584211\">\n<div id=\"fs-id1584212\">\n<p id=\"fs-id1584213\">You decide to paint your kitchen green. You create the color of paint by mixing yellow and blue paints. You cannot remember how many gallons of each color went into your mix, but you know there were 10 gal total. Additionally, you kept your receipt, and know the total amount spent was $29.50. If each gallon of yellow costs $2.59, and each gallon of blue costs $3.19, how many gallons of each color go into your green mix?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1584221\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1584221\"]\n<p id=\"fs-id1584221\">4 gal yellow, 6 gal blue<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1698980\">\n<div id=\"fs-id1698981\">\n<p id=\"fs-id1698982\">You sold two types of scarves at a farmers\u2019 market and would like to know which one was more popular. The total number of scarves sold was 56, the yellow scarf cost $10, and the purple scarf cost $11. If you had total revenue of $583, how many yellow scarves and how many purple scarves were sold?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1698991\">\n<div id=\"fs-id1698992\">\n<p id=\"fs-id1698993\">Your garden produced two types of tomatoes, one green and one red. The red weigh 10 oz, and the green weigh 4 oz. You have 30 tomatoes, and a total weight of 13 lb, 14 oz. How many of each type of tomato do you have?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1699000\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1699000\"]\n<p id=\"fs-id1699000\">13 green tomatoes, 17 red tomatoes<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1648791\">\n<div id=\"fs-id1648792\">\n<p id=\"fs-id1648793\">At a market, the three most popular vegetables make up 53% of vegetable sales. Corn has 4% higher sales than broccoli, which has 5% more sales than onions. What percentage does each vegetable have in the market share?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1648798\">\n<div id=\"fs-id1648799\">\n<p id=\"fs-id1648800\">At the same market, the three most popular fruits make up 37% of the total fruit sold. Strawberries sell twice as much as oranges, and kiwis sell one more percentage point than oranges. For each fruit, find the percentage of total fruit sold.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1648808\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1648808\"]\n<p id=\"fs-id1648808\">Strawberries 18%, oranges 9%, kiwi 10%<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1648813\">\n<div id=\"fs-id1648814\">\n<p id=\"fs-id1648815\">Three bands performed at a concert venue. The first band charged $15 per ticket, the second band charged $45 per ticket, and the final band charged $22 per ticket. There were 510 tickets sold, for a total of $12,700. If the first band had 40 more audience members than the second band, how many tickets were sold for each band?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1637276\">\n<div id=\"fs-id1637277\">\n<p id=\"fs-id1637278\">A movie theatre sold tickets to three movies. The tickets to the first movie were $5, the tickets to the second movie were $11, and the third movie was $12. 100 tickets were sold to the first movie. The total number of tickets sold was 642, for a total revenue of $6,774. How many tickets for each movie were sold?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1637286\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1637286\"]\n<p id=\"fs-id1637286\">100 for movie 1, 230 for movie 2, 312 for movie 3<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1637290\">\n<div id=\"fs-id1637291\">\n<p id=\"fs-id1637292\">Men aged 20\u201329, 30\u201339, and 40\u201349 made up 78% of the population at a prison last year. This year, the same age groups made up 82.08% of the population. The 20\u201329 age group increased by 20%, the 30\u201339 age group increased by 2%, and the 40\u201349 age group decreased to[latex]\\,\\frac{3}{4}\\,[\/latex]of their previous population. Originally, the 30\u201339 age group had 2% more prisoners than the 20\u201329 age group. Determine the prison population percentage for each age group last year.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1425381\">\n<div id=\"fs-id1425382\">\n<p id=\"fs-id1425383\">At a women\u2019s prison down the road, the total number of inmates aged 20\u201349 totaled 5,525. This year, the 20\u201329 age group increased by 10%, the 30\u201339 age group decreased by 20%, and the 40\u201349 age group doubled. There are now 6,040 prisoners. Originally, there were 500 more in the 30\u201339 age group than the 20\u201329 age group. Determine the prison population for each age group last year.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1425394\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1425394\"]\n<p id=\"fs-id1425394\">20\u201329: 2,100, 30\u201339: 2,600, 40\u201349: 825<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1425398\">For the following exercises, use this scenario: A health-conscious company decides to make a trail mix out of almonds, dried cranberries, and chocolate-covered cashews. The nutritional information for these items is shown in <a class=\"autogenerated-content\" href=\"#Table_09_08_01\">(Figure)<\/a>.<\/p>\n\n<table id=\"Table_09_08_01\" summary=\"..\">\n<thead>\n<tr>\n<th><\/th>\n<th>Fat (g)<\/th>\n<th>Protein (g)<\/th>\n<th>Carbohydrates (g)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><strong>Almonds (10)<\/strong><\/td>\n<td>6<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td><strong>Cranberries (10)<\/strong><\/td>\n<td>0.02<\/td>\n<td>0<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td><strong>Cashews (10)<\/strong><\/td>\n<td>7<\/td>\n<td>3.5<\/td>\n<td>5.5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1394923\">\n<div id=\"fs-id1394924\">\n<p id=\"fs-id1394925\">For the special \u201clow-carb\u201dtrail mix, there are 1,000 pieces of mix. The total number of carbohydrates is 425 g, and the total amount of fat is 570.2 g. If there are 200 more pieces of cashews than cranberries, how many of each item is in the trail mix?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1394933\">\n<div id=\"fs-id1394934\">\n<p id=\"fs-id1394935\">For the \u201chiking\u201d mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. If there is the same amount of almonds as cashews, how many of each item is in the trail mix?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1647224\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1647224\"]\n<p id=\"fs-id1647224\">300 almonds, 400 cranberries, 300 cashews<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1647228\">\n<div id=\"fs-id1647230\">\n<p id=\"fs-id1647231\">For the \u201cenergy-booster\u201d mix, there are 1,000 pieces in the mix, containing 145 g of protein and 625 g of carbohydrates. If the number of almonds and cashews summed together is equivalent to the amount of cranberries, how many of each item is in the trail mix?<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"review-exercises textbox exercises\">\n<h3>Review Exercises<\/h3>\n<div id=\"fs-id2876816\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/0bb1dacf-671b-4fe5-a5a4-e67ad9a48d54\">Systems of Linear Equations: Two Variables<\/a><\/h4>\n<p id=\"fs-id1460571\">For the following exercises, determine whether the ordered pair is a solution to the system of equations.<\/p>\n\n<div id=\"fs-id1460575\">\n<div id=\"fs-id1460576\">\n<p id=\"fs-id1460577\">[latex]\\begin{array}{l}3x-y=4\\\\ x+4y=-3\\,\\end{array}[\/latex]and[latex]\\,\\left(-1,1\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1395131\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1395131\"]\n<p id=\"fs-id1395131\">No<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1395135\">\n<div>\n<p id=\"fs-id1395137\">[latex]\\begin{array}{l}6x-2y=24\\\\ -3x+3y=18\\,\\end{array}[\/latex]and[latex]\\,\\left(9,15\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1519982\">For the following exercises, use substitution to solve the system of equations.<\/p>\n\n<div id=\"fs-id1519985\">\n<div id=\"fs-id1519986\">\n<p id=\"fs-id1519987\">[latex]\\begin{array}{l}10x+5y=-5\\hfill \\\\ \\,\\,\\,3x-2y=-12\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1698278\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1698278\"]\n<p id=\"fs-id1698278\">[latex]\\left(-2,3\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1671248\">\n<div id=\"fs-id1671249\">\n<p id=\"fs-id1671250\">[latex]\\begin{array}{l}\\frac{4}{7}x+\\frac{1}{5}y=\\frac{43}{70}\\\\ \\frac{5}{6}x-\\frac{1}{3}y=-\\frac{2}{3}\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1523916\">\n<div id=\"fs-id1523917\">\n<p id=\"fs-id1523918\">[latex]\\begin{array}{l}5x+6y=14\\\\ 4x+8y=8\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1428954\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1428954\"]\n<p id=\"fs-id1428954\">[latex]\\left(4,-1\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1425491\">For the following exercises, use addition to solve the system of equations.<\/p>\n\n<div id=\"fs-id1425494\">\n<div id=\"fs-id1425495\">\n<p id=\"fs-id1425496\">[latex]\\begin{array}{l}3x+2y=-7\\\\ 2x+4y=6\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1512233\">\n<div id=\"fs-id1512234\">\n<p id=\"fs-id1512235\">[latex]\\begin{array}{r}3x+4y=2\\\\ 9x+12y=3\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1584732\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1584732\"]\n<p id=\"fs-id1584732\">No solutions exist.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1584736\">\n<div id=\"fs-id1584737\">\n<p id=\"fs-id1584738\">[latex]\\begin{array}{l}8x+4y=2\\\\ 6x-5y=0.7\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1459641\">For the following exercises, write a system of equations to solve each problem. Solve the system of equations.<\/p>\n\n<div id=\"fs-id1459646\">\n<div id=\"fs-id1459647\">\n<p id=\"fs-id1459648\">A factory has a cost of production[latex]\\,C\\left(x\\right)=150x+15\\text{,}000\\,[\/latex]and a revenue function[latex]\\,R\\left(x\\right)=200x.\\,[\/latex]What is the break-even point?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1536207\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1536207\"]\n<p id=\"fs-id1536207\">[latex]\\left(300,60,000\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1423656\">\n<div id=\"fs-id1423657\">\n<p id=\"fs-id1423658\">A performer charges[latex]\\,C\\left(x\\right)=50x+10\\text{,}000,\\,[\/latex]where[latex]\\,x\\,[\/latex]is the total number of attendees at a show. The venue charges $75 per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1358367\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1358367\"]\n<p id=\"fs-id1358367\">[latex]\\left(400,30,000\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2266991\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/4f2b4755-b59c-4f4f-b4c3-e97938c93fcd\">Systems of Linear Equations: Three Variables<\/a><\/h4>\n<p id=\"fs-id1300053\">For the following exercises, solve the system of three equations using substitution or addition.<\/p>\n\n<div id=\"fs-id1512118\">\n<div id=\"fs-id1512119\">\n<p id=\"fs-id1512120\">[latex]\\begin{array}{l}\\text{ }0.5x-0.5y=10\\hfill \\\\ \\text{ }-0.2y+0.2x=4\\hfill \\\\ \\text{ }0.1x+0.1z=2\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1513414\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1513414\"]\n<p id=\"fs-id1513414\">[latex]\\left(10,-10,10\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1705168\">\n<div id=\"fs-id1705169\">\n<p id=\"fs-id1705170\">[latex]\\begin{array}{r}\\hfill 5x+3y-z=5\\,\\,\\,\\\\ \\hfill 3x-2y+4z=13\\\\ \\hfill 4x+3y+5z=22\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1524026\">\n<div id=\"fs-id1524027\">\n<p id=\"fs-id1524028\">[latex]\\begin{array}{r}x+y+z=1\\\\ 2x+2y+2z=1\\\\ 3x+3y=2\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1366211\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1366211\"]\n<p id=\"fs-id1366211\">No solutions exist.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2633372\">\n<div id=\"fs-id2633373\">\n<p id=\"fs-id2633374\">[latex]\\begin{array}{l}\\text{ }2x-3y+z=-1\\hfill \\\\ \\text{ }x+y+z=-4\\hfill \\\\ \\text{ }4x+2y-3z=33\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1677098\">\n<div id=\"fs-id1677099\">\n<p id=\"fs-id1677100\">[latex]\\begin{array}{l}\\,\\,3x+2y-z=-10\\hfill \\\\ \\,\\,\\,\\,x-y+2z=7\\hfill \\\\ -x+3y+z=-2\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1409562\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1409562\"]\n<p id=\"fs-id1409562\">[latex]\\left(-1,-2,3\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1406449\">\n<div id=\"fs-id1406450\">\n<p id=\"fs-id1406451\">[latex]\\begin{array}{r}\\hfill 3x+4z=-11\\\\ \\hfill x-2y=5\\,\\,\\,\\,\\,\\,\\,\\\\ \\hfill 4y-z=-10\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1417289\">\n<div id=\"fs-id1417290\">\n<p id=\"fs-id1417291\">[latex]\\begin{array}{r}2x-3y+z=0\\\\ 2x+4y-3z=0\\\\ 6x-2y-z=0\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1417064\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1417064\"]\n<p id=\"fs-id1417064\">[latex]\\left(x,\\frac{8x}{5},\\frac{14x}{5}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1532027\">\n<div id=\"fs-id1532028\">\n<p id=\"fs-id1532029\">[latex]\\begin{array}{r}6x-4y-2z=2\\\\ 3x+2y-5z=4\\\\ 6y-7z=5\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1587443\">For the following exercises, write a system of equations to solve each problem. Solve the system of equations.<\/p>\n\n<div id=\"fs-id1587447\">\n<div id=\"fs-id1706452\">\n<p id=\"fs-id1706453\">Three odd numbers sum up to 61. The smaller is one-third the larger and the middle number is 16 less than the larger. What are the three numbers?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1706459\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1706459\"]\n<p id=\"fs-id1706459\">11, 17, 33<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1706464\">\n<div id=\"fs-id1706465\">\n<p id=\"fs-id1706466\">A local theatre sells out for their show. They sell all 500 tickets for a total purse of $8,070.00. The tickets were priced at $15 for students, $12 for children, and $18 for adults. If the band sold three times as many adult tickets as children\u2019s tickets, how many of each type was sold?<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1801019\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/228bf1c6-e200-41b3-9bc0-79b6f6edbb37\">Systems of Nonlinear Equations and Inequalities: Two Variables<\/a><\/h4>\n<p id=\"fs-id1706479\">For the following exercises, solve the system of nonlinear equations.<\/p>\n\n<div id=\"fs-id1584246\">\n<div id=\"fs-id1584247\">\n<p id=\"fs-id1584248\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ y={x}^{2}-7\\end{array}\\hfill \\\\ y=5x-13\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1695047\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1695047\"]\n<p id=\"fs-id1695047\">[latex]\\left(2,-3\\right),\\left(3,2\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1403228\">\n<div id=\"fs-id1403229\">\n<p id=\"fs-id1403230\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ y={x}^{2}-4\\end{array}\\hfill \\\\ y=5x+10\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1434420\">\n<div id=\"fs-id1434421\">\n<p id=\"fs-id1434422\">[latex]\\begin{array}{l}{x}^{2}+{y}^{2}=16\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y=x-8\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1403141\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1403141\"]\n<p id=\"fs-id1403141\">No solution<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1403146\">\n<div id=\"fs-id1403147\">\n<p id=\"fs-id1403148\">[latex]\\begin{array}{l}{x}^{2}+{y}^{2}=25\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y={x}^{2}+5\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1451621\">\n<div id=\"fs-id1451622\">\n<p id=\"fs-id1451623\">[latex]\\begin{array}{r}{x}^{2}+{y}^{2}=4\\\\ y-{x}^{2}=3\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1513095\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1513095\"]\n<p id=\"fs-id1513095\">No solution<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1513100\">For the following exercises, graph the inequality.<\/p>\n\n<div id=\"fs-id1513103\">\n<div id=\"fs-id1513104\">\n<p id=\"fs-id1513105\">[latex]y&gt;{x}^{2}-1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1457141\">\n<div id=\"fs-id1457142\">\n<p id=\"fs-id1457143\">[latex]\\frac{1}{4}{x}^{2}+{y}^{2}&lt;4[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"1663822\"]Show Solution[\/reveal-answer][hidden-answer a=\"1663822\"]<span id=\"fs-id1663829\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154215\/CNX_Precalc_Figure_09_08_202.jpg\" alt=\"\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<p id=\"fs-id1663839\">For the following exercises, graph the system of inequalities.<\/p>\n\n<div id=\"fs-id1663842\">\n<div id=\"fs-id1663844\">\n<p id=\"fs-id1663845\">[latex]\\begin{array}{l}{x}^{2}+{y}^{2}+2x&lt;3\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y&gt;-{x}^{2}-3\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1646717\">\n<div id=\"fs-id1646718\">\n<p id=\"fs-id1646719\">[latex]\\begin{array}{l}{x}^{2}-2x+{y}^{2}-4x&lt;4\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y&lt;-x+4\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"1600811\"]Show Solution[\/reveal-answer][hidden-answer a=\"1600811\"]<span id=\"fs-id1600817\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154218\/CNX_Precalc_Figure_09_08_204.jpg\" alt=\"\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1600828\">\n<div id=\"fs-id1600829\">\n<p id=\"fs-id1600830\">[latex]\\begin{array}{l}{x}^{2}+{y}^{2}&lt;1\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{y}^{2}&lt;x\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2867422\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/b61e8ccf-4f28-453e-9e41-99b6b7c8dfe8\">Partial Fractions<\/a><\/h4>\n<p id=\"fs-id1409014\">For the following exercises, decompose into partial fractions.<\/p>\n\n<div>\n<div id=\"fs-id1409018\">\n<p id=\"fs-id1409019\">[latex]\\frac{-2x+6}{{x}^{2}+3x+2}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1530172\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1530172\"]\n<p id=\"fs-id1530172\">[latex]\\frac{2}{x+2},\\frac{-4}{x+1}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1430683\">\n<div id=\"fs-id1430684\">\n<p id=\"fs-id1430685\">[latex]\\frac{10x+2}{4{x}^{2}+4x+1}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1564134\">\n<div id=\"fs-id1564135\">\n<p id=\"fs-id1564136\">[latex]\\frac{7x+20}{{x}^{2}+10x+25}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1440535\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1440535\"]\n<p id=\"fs-id1440535\">[latex]\\frac{7}{x+5},\\frac{-15}{{\\left(x+5\\right)}^{2}}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1614973\">\n<div id=\"fs-id1614974\">\n<p id=\"fs-id1614975\">[latex]\\frac{x-18}{{x}^{2}-12x+36}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1417354\">\n<div id=\"fs-id1417355\">\n<p id=\"fs-id1417356\">[latex]\\frac{-{x}^{2}+36x+70}{{x}^{3}-125}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1528465\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1528465\"]\n<p id=\"fs-id1528465\">[latex]\\frac{3}{x-5},\\frac{-4x+1}{{x}^{2}+5x+25}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1583086\">\n<div id=\"fs-id1583087\">\n<p id=\"fs-id1583088\">[latex]\\frac{-5{x}^{2}+6x-2}{{x}^{3}+27}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1431813\">\n<div id=\"fs-id1431814\">\n<p id=\"fs-id1431815\">[latex]\\frac{{x}^{3}-4{x}^{2}+3x+11}{{\\left({x}^{2}-2\\right)}^{2}}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1582588\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1582588\"]\n<p id=\"fs-id1582588\">[latex]\\frac{x-4}{\\left({x}^{2}-2\\right)},\\frac{5x+3}{{\\left({x}^{2}-2\\right)}^{2}}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1655916\">\n<div id=\"fs-id1655917\">\n<p id=\"fs-id1655918\">[latex]\\frac{4{x}^{4}-2{x}^{3}+22{x}^{2}-6x+48}{x{\\left({x}^{2}+4\\right)}^{2}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2925191\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/508a4d4e-0136-4c6c-87b8-e210022d69b4\">Matrices and Matrix Operations<\/a><\/h4>\n<p id=\"fs-id1581865\">For the following exercises, perform the requested operations on the given matrices.<\/p>\n\n<div id=\"fs-id1581869\">[latex]A=\\left[\\begin{array}{rr}\\hfill 4&amp; \\hfill -2\\\\ \\hfill 1&amp; \\hfill 3\\end{array}\\right],B=\\left[\\begin{array}{rrr}\\hfill 6&amp; \\hfill 7&amp; \\hfill -3\\\\ \\hfill 11&amp; \\hfill -2&amp; \\hfill 4\\end{array}\\right],C=\\left[\\begin{array}{r}\\hfill \\begin{array}{cc}6&amp; 7\\\\ 11&amp; -2\\end{array}\\\\ \\hfill \\begin{array}{cc}14&amp; 0\\end{array}\\end{array}\\right],D=\\left[\\begin{array}{rrr}\\hfill 1&amp; \\hfill -4&amp; \\hfill 9\\\\ \\hfill 10&amp; \\hfill 5&amp; \\hfill -7\\\\ \\hfill 2&amp; \\hfill 8&amp; \\hfill 5\\end{array}\\right],E=\\left[\\begin{array}{rrr}\\hfill 7&amp; \\hfill -14&amp; \\hfill 3\\\\ \\hfill 2&amp; \\hfill -1&amp; \\hfill 3\\\\ \\hfill 0&amp; \\hfill 1&amp; \\hfill 9\\end{array}\\right][\/latex]<\/div>\n<div id=\"fs-id1519664\">\n<div id=\"fs-id1519665\">\n<p id=\"fs-id1519666\">[latex]-4A[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1615553\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1615553\"]\n<p id=\"fs-id1615553\">[latex]\\left[\\begin{array}{cc}-16&amp; 8\\\\ -4&amp; -12\\end{array}\\right][\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1535716\">\n<div id=\"fs-id1535717\">\n<p id=\"fs-id1535718\">[latex]10D-6E[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1536793\">\n<div id=\"fs-id1536794\">\n<p id=\"fs-id1536796\">[latex]B+C[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1706172\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1706172\"]\n<p id=\"fs-id1706172\">undefined; dimensions do not match<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1706176\">\n<div id=\"fs-id1706177\">\n<p id=\"fs-id1706178\">[latex]AB[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1650861\">\n<div id=\"fs-id1650862\">\n<p id=\"fs-id1650863\">[latex]BA[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1650879\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1650879\"]\n<p id=\"fs-id1650879\">undefined; inner dimensions do not match<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1650884\">\n<div id=\"fs-id1650885\">\n<p id=\"fs-id1650886\">[latex]BC[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1647717\">\n<div id=\"fs-id1647718\">\n<p id=\"fs-id1647720\">[latex]CB[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1647736\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1647736\"]\n<p id=\"fs-id1647736\">[latex]\\left[\\begin{array}{ccc}113&amp; 28&amp; 10\\\\ 44&amp; 81&amp; -41\\\\ 84&amp; 98&amp; -42\\end{array}\\right][\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1419710\">\n<div id=\"fs-id1419711\">\n<p id=\"fs-id1419712\">[latex]DE[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1419726\">\n<div id=\"fs-id1419727\">\n<p id=\"fs-id1419728\">[latex]ED[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1693755\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1693755\"]\n<p id=\"fs-id1693755\">[latex]\\left[\\begin{array}{ccc}-127&amp; -74&amp; 176\\\\ -2&amp; 11&amp; 40\\\\ 28&amp; 77&amp; 38\\end{array}\\right][\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1366518\">\n<div id=\"fs-id1366519\">\n<p id=\"fs-id1366520\">[latex]EC[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1366534\">\n<div id=\"fs-id1366535\">\n<p id=\"fs-id1366536\">[latex]CE[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1700302\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1700302\"]\n<p id=\"fs-id1700302\">undefined; inner dimensions do not match<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1700306\">\n<div id=\"fs-id1700307\">\n<p id=\"fs-id1700308\">[latex]{A}^{3}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2652002\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/69e0b9a7-0928-46a6-bb59-6cd28f18eec9\">Solving Systems with Gaussian Elimination<\/a><\/h4>\n<p id=\"fs-id1407521\">For the following exercises, write the system of linear equations from the augmented matrix. Indicate whether there will be a unique solution.<\/p>\n\n<div id=\"fs-id1407525\">\n<div id=\"fs-id1407526\">\n<p id=\"fs-id1407528\">[latex]\\left[\\begin{array}{rrr}\\hfill 1&amp; \\hfill 0&amp; \\hfill -3\\\\ \\hfill 0&amp; \\hfill 1&amp; \\hfill 2\\\\ \\hfill 0&amp; \\hfill 0&amp; \\hfill 0\\end{array}\\text{ }|\\text{ }\\begin{array}{r}\\hfill 7\\\\ \\hfill -5\\\\ \\hfill 0\\end{array}\\right][\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1637532\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1637532\"]\n<p id=\"fs-id1637532\">[latex]\\begin{array}{l}x-3z=7\\\\ y+2z=-5\\,\\end{array}[\/latex]with infinite solutions<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1523479\">\n<div>\n<p id=\"fs-id1523481\">[latex]\\left[\\begin{array}{rrr}\\hfill 1&amp; \\hfill 0&amp; \\hfill 5\\\\ \\hfill 0&amp; \\hfill 1&amp; \\hfill -2\\\\ \\hfill 0&amp; \\hfill 0&amp; \\hfill 0\\end{array}\\text{ }|\\text{ }\\begin{array}{r}\\hfill -9\\\\ \\hfill 4\\\\ \\hfill 3\\end{array}\\right][\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1375129\">For the following exercises, write the augmented matrix from the system of linear equations.<\/p>\n\n<div id=\"fs-id1375132\">\n<div id=\"fs-id1375133\">\n<p id=\"fs-id1375134\">[latex]\\begin{array}{l}\\\\ \\begin{array}{r}\\hfill -2x+2y+z=7\\\\ \\hfill 2x-8y+5z=0\\\\ \\hfill 19x-10y+22z=3\\end{array}\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1656057\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1656057\"]\n<p id=\"fs-id1656057\">[latex]\\left[\\begin{array}{rrr}\\hfill -2&amp; \\hfill 2&amp; \\hfill 1\\\\ \\hfill 2&amp; \\hfill -8&amp; \\hfill 5\\\\ \\hfill 19&amp; \\hfill -10&amp; \\hfill 22\\end{array}\\text{ }|\\text{ }\\begin{array}{r}\\hfill 7\\\\ \\hfill 0\\\\ \\hfill 3\\end{array}\\right][\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1452352\">\n<div id=\"fs-id1452353\">\n<p id=\"fs-id1452354\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,4x+2y-3z=14\\hfill \\\\ -12x+3y+z=100\\hfill \\\\ \\,\\,\\,\\,\\,9x-6y+2z=31\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1457569\">\n<div id=\"fs-id1457570\">\n<p id=\"fs-id1457571\">[latex]\\begin{array}{r}\\hfill x+3z=12\\,\\\\ \\hfill -x+4y=0\\,\\,\\,\\,\\\\ \\hfill y+2z=-7\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1528512\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1528512\"]\n<p id=\"fs-id1528512\">[latex]\\left[\\begin{array}{rrr}\\hfill 1&amp; \\hfill 0&amp; \\hfill 3\\\\ \\hfill -1&amp; \\hfill 4&amp; \\hfill 0\\\\ \\hfill 0&amp; \\hfill 1&amp; \\hfill 2\\end{array}\\text{ }|\\text{ }\\begin{array}{r}\\hfill 12\\\\ \\hfill 0\\\\ \\hfill -7\\end{array}\\right][\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1601028\">For the following exercises, solve the system of linear equations using Gaussian elimination.<\/p>\n\n<div id=\"fs-id1410109\">\n<div id=\"fs-id1410110\">\n<p id=\"fs-id1410111\">[latex]\\begin{array}{r}3x-4y=-7\\\\ -6x+8y=14\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1626545\">\n<div id=\"fs-id1626546\">\n<p id=\"fs-id1626548\">[latex]\\begin{array}{r}3x-4y=1\\\\ -6x+8y=6\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1422121\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1422121\"]\n<p id=\"fs-id1422121\">No solutions exist.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1503778\">\n<div id=\"fs-id1503779\">\n<p id=\"fs-id1503780\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ -1.1x-2.3y=6.2\\end{array}\\hfill \\\\ -5.2x-4.1y=4.3\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1436195\">\n<div id=\"fs-id1436196\">\n<p id=\"fs-id1436197\">[latex]\\begin{array}{r}\\hfill 2x+3y+2z=1\\,\\,\\,\\,\\,\\\\ \\hfill -4x-6y-4z=-2\\\\ \\hfill 10x+15y+10z=0\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1455902\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1455902\"]\n<p id=\"fs-id1455902\">No solutions exist.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1455907\">\n<div id=\"fs-id1455908\">\n<p id=\"fs-id1455909\">[latex]\\begin{array}{r}\\hfill -x+2y-4z=8\\,\\,\\,\\,\\\\ \\hfill 3y+8z=-4\\\\ \\hfill -7x+y+2z=1\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1766016\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/634d2387-7429-480f-a04e-867c2f9699fb\">Solving Systems with Inverses<\/a><\/h4>\n<p id=\"fs-id1431604\">For the following exercises, find the inverse of the matrix.<\/p>\n\n<div id=\"fs-id1431608\">\n<div id=\"fs-id1431609\">\n<p id=\"fs-id1431610\">[latex]\\left[\\begin{array}{rr}\\hfill -0.2&amp; \\hfill 1.4\\\\ \\hfill 1.2&amp; \\hfill -0.4\\end{array}\\right][\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1531734\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1531734\"]\n<p id=\"fs-id1531734\">[latex]\\frac{1}{8}\\left[\\begin{array}{cc}2&amp; 7\\\\ 6&amp; 1\\end{array}\\right][\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1949652\">\n<div id=\"fs-id1949653\">\n<p id=\"fs-id1949654\">[latex]\\left[\\begin{array}{rr}\\hfill \\frac{1}{2}&amp; \\hfill -\\frac{1}{2}\\\\ \\hfill -\\frac{1}{4}&amp; \\hfill \\frac{3}{4}\\end{array}\\right][\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1456514\">\n<div id=\"fs-id1456515\">\n<p id=\"fs-id1456516\">[latex]\\left[\\begin{array}{ccc}12&amp; 9&amp; -6\\\\ -1&amp; 3&amp; 2\\\\ -4&amp; -3&amp; 2\\end{array}\\right][\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1514199\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1514199\"]\n<p id=\"fs-id1514199\">No inverse exists.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1514203\">\n<div id=\"fs-id1514204\">\n<p id=\"fs-id1514205\">[latex]\\left[\\begin{array}{ccc}2&amp; 1&amp; 3\\\\ 1&amp; 2&amp; 3\\\\ 3&amp; 2&amp; 1\\end{array}\\right][\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1507401\">For the following exercises, find the solutions by computing the inverse of the matrix.<\/p>\n\n<div id=\"fs-id1507405\">\n<div id=\"fs-id1507406\">\n<p id=\"fs-id1507407\">[latex]\\begin{array}{l}\\,\\,\\,\\,0.3x-0.1y=-10\\hfill \\\\ -0.1x+0.3y=14\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id299169\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id299169\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id299169\"]\n<p id=\"fs-id299171\">[latex]\\left(-20,40\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1601516\">\n<div id=\"fs-id1601518\">\n<p id=\"fs-id1601519\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,0.4x-0.2y=-0.6\\hfill \\\\ -0.1x+0.05y=0.3\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1584782\">\n<div id=\"fs-id1584783\">\n<p id=\"fs-id1584784\">[latex]\\begin{array}{r}4x+3y-3z=-4.3\\\\ 5x-4y-z=-6.1\\\\ x+z=-0.7\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1531936\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1531936\"]\n<p id=\"fs-id1531936\">[latex]\\left(-1,0.2,0.3\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1519539\">\n<div id=\"fs-id1519540\">\n<p id=\"fs-id1519542\">[latex]\\begin{array}{r}\\hfill \\begin{array}{l}\\\\ -2x-3y+2z=3\\end{array}\\\\ \\hfill -x+2y+4z=-5\\\\ \\hfill -2y+5z=-3\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1403397\">For the following exercises, write a system of equations to solve each problem. Solve the system of equations.<\/p>\n\n<div id=\"fs-id1403401\">\n<div id=\"fs-id1403402\">\n<p id=\"fs-id1403403\">Students were asked to bring their favorite fruit to class. 90% of the fruits consisted of banana, apple, and oranges. If oranges were half as popular as bananas and apples were 5% more popular than bananas, what are the percentages of each individual fruit?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1403411\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1403411\"]\n<p id=\"fs-id1403411\">17% oranges, 34% bananas, 39% apples<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1403415\">\n<div id=\"fs-id1403416\">\n<p id=\"fs-id1403417\">A sorority held a bake sale to raise money and sold brownies and chocolate chip cookies. They priced the brownies at $2 and the chocolate chip cookies at $1. They raised $250 and sold 175 items. How many brownies and how many cookies were sold?<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2974900\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/e75dadb0-5d43-42f2-8e78-0ad237f10096\">Solving Systems with Cramer's Rule<\/a><\/h4>\n<p id=\"fs-id1357039\">For the following exercises, find the determinant.<\/p>\n\n<div id=\"fs-id1357042\">\n<div id=\"fs-id1357043\">\n<p id=\"fs-id1357044\">[latex]|\\begin{array}{cc}100&amp; 0\\\\ 0&amp; 0\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1504600\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1504600\"]\n<p id=\"fs-id1504600\">0<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1504604\">\n<div id=\"fs-id1504605\">\n<p id=\"fs-id1504606\">[latex]|\\begin{array}{cc}0.2&amp; -0.6\\\\ 0.7&amp; -1.1\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1535261\">\n<div id=\"fs-id1535262\">\n<p id=\"fs-id1535263\">[latex]|\\begin{array}{ccc}-1&amp; 4&amp; 3\\\\ 0&amp; 2&amp; 3\\\\ 0&amp; 0&amp; -3\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1698281\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1698281\"]\n<p id=\"fs-id1698281\">6<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1698286\">\n<div id=\"fs-id1698287\">\n<p id=\"fs-id1698288\">[latex]|\\begin{array}{ccc}\\sqrt{2}&amp; 0&amp; 0\\\\ 0&amp; \\sqrt{2}&amp; 0\\\\ 0&amp; 0&amp; \\sqrt{2}\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1534786\">For the following exercises, use Cramer\u2019s Rule to solve the linear systems of equations.<\/p>\n\n<div id=\"fs-id1534791\">\n<div id=\"fs-id1534792\">\n<p id=\"fs-id1534793\">[latex]\\begin{array}{r}\\hfill 4x-2y=23\\,\\,\\,\\,\\\\ \\hfill -5x-10y=-35\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1508832\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1508832\"]\n<p id=\"fs-id1508832\">[latex]\\left(6,\\frac{1}{2}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1698540\">\n<div id=\"fs-id1698541\">\n<p id=\"fs-id1698542\">[latex]\\begin{array}{l}0.2x-0.1y=0\\\\ -0.3x+0.3y=2.5\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1689845\">\n<div id=\"fs-id1689846\">\n<p id=\"fs-id1689847\">[latex]\\begin{array}{r}\\hfill -0.5x+0.1y=0.3\\,\\,\\,\\\\ \\hfill -0.25x+0.05y=0.15\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1584180\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1584180\"]\n<p id=\"fs-id1584180\">(<em>x<\/em>, 5<em>x <\/em>+ 3)<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1584193\">\n<div id=\"fs-id1508743\">\n<p id=\"fs-id1508744\">[latex]\\begin{array}{l}x+6y+3z=4\\\\ 2x+y+2z=3\\\\ 3x-2y+z=0\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1676826\">\n<div id=\"fs-id1676827\">\n<p id=\"fs-id1676828\">[latex]\\begin{array}{r}\\hfill 4x-3y+5z=-\\frac{5}{2}\\\\ \\hfill 7x-9y-3z=\\frac{3}{2}\\,\\,\\,\\,\\\\ \\hfill x-5y-5z=\\frac{5}{2}\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1508865\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1508865\"]\n<p id=\"fs-id1508865\">[latex]\\left(0,0,-\\frac{1}{2}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1658138\">\n<div id=\"fs-id1658139\">\n<p id=\"fs-id1658140\">[latex]\\begin{array}{r}\\frac{3}{10}x-\\frac{1}{5}y-\\frac{3}{10}z=-\\frac{1}{50}\\\\ \\frac{1}{10}x-\\frac{1}{10}y-\\frac{1}{2}z=-\\frac{9}{50}\\\\ \\frac{2}{5}x-\\frac{1}{2}y-\\frac{3}{5}z=-\\frac{1}{5}\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1459744\" class=\"practice-test\">\n<h3>Practice Test<\/h3>\n<p id=\"fs-id1459748\">Is the following ordered pair a solution to the system of equations?<\/p>\n\n<div id=\"fs-id1459751\">\n<div id=\"fs-id1459752\">\n<p id=\"fs-id1459753\">[latex]\\begin{array}{l}\\\\ \\begin{array}{l}-5x-y=12\\,\\hfill \\\\ x+4y=9\\hfill \\end{array}\\end{array}[\/latex]with[latex]\\,\\left(-3,3\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1435942\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1435942\"]\n<p id=\"fs-id1435942\">Yes<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1403240\">For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.<\/p>\n\n<div id=\"fs-id1403244\">\n<div id=\"fs-id1403246\">\n<p id=\"fs-id1403247\">[latex]\\begin{array}{r}\\frac{1}{2}x-\\frac{1}{3}y=4\\\\ \\frac{3}{2}x-y=0\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1406347\">\n<div id=\"fs-id1406348\">\n<p id=\"fs-id1406349\">[latex]\\begin{array}{r}\\hfill \\begin{array}{l}\\\\ -\\frac{1}{2}x-4y=4\\end{array}\\\\ \\hfill 2x+16y=2\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1406246\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1406246\"]\n<p id=\"fs-id1406246\">No solutions exist.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1406251\">\n<div id=\"fs-id1406252\">\n<p id=\"fs-id1406253\">[latex]\\begin{array}{r}\\hfill 5x-y=1\\,\\,\\,\\,\\\\ \\hfill -10x+2y=-2\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1780639\">\n<div id=\"fs-id1780640\">\n<p id=\"fs-id1780641\">[latex]\\begin{array}{l}4x-6y-2z=\\frac{1}{10}\\hfill \\\\ \\,\\,\\,x-7y+5z=-\\frac{1}{4}\\hfill \\\\ 3x+6y-9z=\\frac{6}{5}\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1455840\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1455840\"]\n<p id=\"fs-id1455840\">[latex]\\frac{1}{20}\\left(10,5,4\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1675200\">\n<div id=\"fs-id1675201\">\n<p id=\"fs-id1675202\">[latex]\\begin{array}{r}x+z=20\\\\ x+y+z=20\\\\ x+2y+z=10\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1436662\">\n<div id=\"fs-id1436663\">\n<p id=\"fs-id1436664\">[latex]\\begin{array}{r}5x-4y-3z=0\\\\ 2x+y+2z=0\\\\ x-6y-7z=0\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1615810\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1615810\"]\n<p id=\"fs-id1615810\">[latex]\\left(x,\\frac{16x}{5}-\\frac{13x}{5}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1537027\">\n<div id=\"fs-id1537028\">\n<p id=\"fs-id1537029\">[latex]\\begin{array}{l}y={x}^{2}+2x-3\\\\ y=x-1\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1432229\">\n<div id=\"fs-id1432230\">\n<p id=\"fs-id1432231\">[latex]\\begin{array}{l}{y}^{2}+{x}^{2}=25\\\\ {y}^{2}-2{x}^{2}=1\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1703248\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1703248\"]\n<p id=\"fs-id1703248\">[latex]\\left(-2\\sqrt{2},-\\sqrt{17}\\right),\\left(-2\\sqrt{2},\\sqrt{17}\\right),\\left(2\\sqrt{2},-\\sqrt{17}\\right),\\left(2\\sqrt{2},\\sqrt{17}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1384560\">For the following exercises, graph the following inequalities.<\/p>\n\n<div id=\"fs-id1384563\">\n<div id=\"fs-id1384564\">\n<p id=\"fs-id1384565\">[latex]y&lt;{x}^{2}+9[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1536358\">\n<div id=\"fs-id1536359\">\n<p id=\"fs-id1536360\">[latex]\\begin{array}{l}{x}^{2}+{y}^{2}&gt;4\\\\ y&lt;{x}^{2}+1\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"1380134\"]Show Solution[\/reveal-answer][hidden-answer a=\"1380134\"]<span id=\"fs-id1380141\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154228\/CNX_Precalc_Figure_09_08_207.jpg\" alt=\"\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<p id=\"fs-id1663752\">For the following exercises, write the partial fraction decomposition.<\/p>\n\n<div id=\"fs-id1663755\">\n<div id=\"fs-id1663756\">\n<p id=\"fs-id1663757\">[latex]\\frac{-8x-30}{{x}^{2}+10x+25}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1507242\">\n<div id=\"fs-id1507243\">\n<p id=\"fs-id1598618\">[latex]\\frac{13x+2}{{\\left(3x+1\\right)}^{2}}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1519238\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1519238\"]\n<p id=\"fs-id1519238\">[latex]\\frac{5}{3x+1}-\\frac{2x+3}{{\\left(3x+1\\right)}^{2}}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1698396\">\n<div id=\"fs-id1698397\">\n<p id=\"fs-id1698398\">[latex]\\frac{{x}^{4}-{x}^{3}+2x-1}{x{\\left({x}^{2}+1\\right)}^{2}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1675019\">For the following exercises, perform the given matrix operations.<\/p>\n\n<div id=\"fs-id1675022\">\n<div id=\"fs-id1675024\">\n<p id=\"fs-id1675025\">[latex]5\\left[\\begin{array}{cc}4&amp; 9\\\\ -2&amp; 3\\end{array}\\right]+\\frac{1}{2}\\left[\\begin{array}{cc}-6&amp; 12\\\\ 4&amp; -8\\end{array}\\right][\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1694107\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1694107\"]\n<p id=\"fs-id1694107\">[latex]\\left[\\begin{array}{cc}17&amp; 51\\\\ -8&amp; 11\\end{array}\\right][\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1527496\">\n<div id=\"fs-id1527497\">\n<p id=\"fs-id1527498\">[latex]\\left[\\begin{array}{rrr}\\hfill 1&amp; \\hfill 4&amp; \\hfill -7\\\\ \\hfill -2&amp; \\hfill 9&amp; \\hfill 5\\\\ \\hfill 12&amp; \\hfill 0&amp; \\hfill -4\\end{array}\\right]\\text{ }\\left[\\begin{array}{cc}3&amp; -4\\\\ 1&amp; 3\\\\ 5&amp; 10\\end{array}\\right][\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1392877\">\n<div id=\"fs-id1392878\">\n<p id=\"fs-id1392879\">[latex]{\\left[\\begin{array}{rr}\\hfill \\frac{1}{2}&amp; \\hfill \\frac{1}{3}\\\\ \\hfill \\frac{1}{4}&amp; \\hfill \\frac{1}{5}\\end{array}\\right]}^{-1}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1583988\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1583988\"]\n<p id=\"fs-id1583988\">[latex]\\left[\\begin{array}{cc}12&amp; -20\\\\ -15&amp; 30\\end{array}\\right][\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1644580\">\n<div id=\"fs-id1644581\">\n<p id=\"fs-id1644582\">[latex]\\mathrm{det}|\\begin{array}{cc}0&amp; 0\\\\ 400&amp; 4\\text{,}000\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1648258\">\n<div id=\"fs-id1648259\">\n<p id=\"fs-id1648260\">[latex]\\mathrm{det}|\\begin{array}{rrr}\\hfill \\frac{1}{2}&amp; \\hfill -\\frac{1}{2}&amp; \\hfill 0\\\\ \\hfill -\\frac{1}{2}&amp; \\hfill 0&amp; \\hfill \\frac{1}{2}\\\\ \\hfill 0&amp; \\hfill \\frac{1}{2}&amp; \\hfill 0\\end{array}|[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1295612\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1295612\"]\n<p id=\"fs-id1295612\">[latex]-\\frac{1}{8}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1615192\">\n<div id=\"fs-id1615193\">\n<p id=\"fs-id1615194\">If[latex]\\,\\mathrm{det}\\left(A\\right)=-6,\\,[\/latex] what would be the determinant if you switched rows 1 and 3, multiplied the second row by 12, and took the inverse?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1595084\">\n<div id=\"fs-id1595085\">\n<p id=\"fs-id1595086\">Rewrite the system of linear equations as an augmented matrix.<\/p>\n\n<div id=\"fs-id1595090\">[latex]\\begin{array}{l}14x-2y+13z=140\\hfill \\\\ -2x+3y-6z=-1\\hfill \\\\ x-5y+12z=11\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1456561\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1456561\"]\n<p id=\"fs-id1456561\">[latex]\\left[\\begin{array}{rrr}\\hfill 14&amp; \\hfill -2&amp; \\hfill 13\\\\ \\hfill -2&amp; \\hfill 3&amp; \\hfill -6\\\\ \\hfill 1&amp; \\hfill -5&amp; \\hfill 12\\end{array}\\text{ }|\\text{ }\\begin{array}{r}\\hfill 140\\\\ \\hfill -1\\\\ \\hfill 11\\end{array}\\right][\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1599476\">\n<div id=\"fs-id1599477\">\n<p id=\"fs-id1599478\">Rewrite the augmented matrix as a system of linear equations.<\/p>\n\n<div id=\"fs-id1599481\">[latex]\\left[\\begin{array}{rrr}\\hfill 1&amp; \\hfill 0&amp; \\hfill 3\\\\ \\hfill -2&amp; \\hfill 4&amp; \\hfill 9\\\\ \\hfill -6&amp; \\hfill 1&amp; \\hfill 2\\end{array}|\\begin{array}{r}\\hfill 12\\\\ \\hfill -5\\\\ \\hfill 8\\end{array}\\right][\/latex]<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1636943\">For the following exercises, use Gaussian elimination to solve the systems of equations.<\/p>\n\n<div id=\"fs-id1636946\">\n<div id=\"fs-id1636947\">\n<p id=\"fs-id1636948\">[latex]\\begin{array}{r}x-6y=4\\\\ 2x-12y=0\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1518911\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1518911\"]\n<p id=\"fs-id1518911\">No solutions exist.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1518915\">\n<div id=\"fs-id1518916\">\n<p id=\"fs-id1518917\">[latex]\\begin{array}{r}\\hfill 2x+y+z=-3\\\\ \\hfill x-2y+3z=6\\,\\,\\,\\,\\\\ \\hfill x-y-z=6\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1699601\">For the following exercises, use the inverse of a matrix to solve the systems of equations.<\/p>\n\n<div id=\"fs-id1699604\">\n<div id=\"fs-id1699605\">\n<p id=\"fs-id1699606\">[latex]\\begin{array}{r}\\hfill 4x-5y=-50\\\\ \\hfill -x+2y=80\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1440484\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1440484\"]\n<p id=\"fs-id1440484\">[latex]\\left(100,90\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1568663\">\n<div id=\"fs-id1568664\">\n<p id=\"fs-id1568666\">[latex]\\begin{array}{r}\\hfill \\frac{1}{100}x-\\frac{3}{100}y+\\frac{1}{20}z=-49\\\\ \\hfill \\frac{3}{100}x-\\frac{7}{100}y-\\frac{1}{100}z=13\\,\\,\\,\\,\\\\ \\hfill \\frac{9}{100}x-\\frac{9}{100}y-\\frac{9}{100}z=99\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1653924\">For the following exercises, use Cramer\u2019s Rule to solve the systems of equations.<\/p>\n\n<div id=\"fs-id1653928\">\n<div id=\"fs-id1653929\">\n<p id=\"fs-id1653930\">[latex]\\begin{array}{l}200x-300y=2\\\\ 400x+715y=4\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1703905\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1703905\"]\n<p id=\"fs-id1703905\">[latex]\\left(\\frac{1}{100},0\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1527386\">\n<div id=\"fs-id1527387\">\n<p id=\"fs-id1527388\">[latex]\\begin{array}{l}0.1x+0.1y-0.1z=-1.2\\\\ 0.1x-0.2y+0.4z=-1.2\\\\ 0.5x-0.3y+0.8z=-5.9\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1528262\">For the following exercises, solve using a system of linear equations.<\/p>\n\n<div id=\"fs-id1528266\">\n<div id=\"fs-id1528267\">\n<p id=\"fs-id1528268\">A factory producing cell phones has the following cost and revenue functions:[latex]\\,C\\left(x\\right)={x}^{2}+75x+2\\text{,}688\\,[\/latex]and[latex]\\,R\\left(x\\right)={x}^{2}+160x.\\,[\/latex]What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1598123\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1598123\"]\n<p id=\"fs-id1598123\">32 or more cell phones per day<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1598127\">\n<div id=\"fs-id1598128\">\n<p id=\"fs-id1598129\">A small fair charges $1.50 for students, $1 for children, and $2 for adults. In one day, three times as many children as adults attended. A total of 800 tickets were sold for a total revenue of $1,050. How many of each type of ticket was sold?<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1674058\">\n \t<dt>Cramer\u2019s Rule<\/dt>\n \t<dd id=\"fs-id1674063\">a method for solving systems of equations that have the same number of equations as variables using determinants<\/dd>\n<\/dl>\n<dl id=\"fs-id1674068\">\n \t<dt>determinant<\/dt>\n \t<dd id=\"fs-id1674074\">a number calculated using the entries of a square matrix that determines such information as whether there is a solution to a system of equations<\/dd>\n<\/dl>\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Evaluate \u20092 \u00d7 2\u2009 determinants.<\/li>\n<li>Use Cramer\u2019s Rule to solve a system of equations in two variables.<\/li>\n<li>Evaluate \u20093 \u00d7 3\u2009 determinants.<\/li>\n<li>Use Cramer\u2019s Rule to solve a system of three equations in three variables.<\/li>\n<li>Know the properties of determinants.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1705215\">We have learned how to solve systems of equations in two variables and three variables, and by multiple methods: substitution, addition, Gaussian elimination, using the inverse of a matrix, and graphing. Some of these methods are easier to apply than others and are more appropriate in certain situations. In this section, we will study two more strategies for solving systems of equations.<\/p>\n<div id=\"fs-id1423475\" class=\"bc-section section\">\n<h3>Evaluating the Determinant of a 2\u00d72 Matrix<\/h3>\n<p id=\"fs-id1331675\">A determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a <span class=\"no-emphasis\">square matrix<\/span> to determine whether there is a solution to the system of equations. Perhaps one of the more interesting applications, however, is their use in cryptography. Secure signals or messages are sometimes sent encoded in a matrix. The data can only be decrypted with an <span class=\"no-emphasis\">invertible matrix<\/span> and the determinant. For our purposes, we focus on the determinant as an indication of the invertibility of the matrix. Calculating the determinant of a matrix involves following the specific patterns that are outlined in this section.<\/p>\n<div id=\"fs-id1305022\">\n<h3>Find the Determinant of a 2 \u00d7 2 Matrix<\/h3>\n<p id=\"fs-id1376906\">The determinant of a[latex]\\,2\\text{ }\u00d7\\text{ }2\\,[\/latex]matrix, given<\/p>\n<div id=\"fs-id1355183\" class=\"unnumbered aligncenter\">[latex]A=\\left[\\begin{array}{cc}a& b\\\\ c& d\\end{array}\\right][\/latex]<\/div>\n<p id=\"fs-id1644238\">is defined as<\/p>\n<p><span id=\"fs-id1347161\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154150\/CNX_Precalc_Figure_09_08_001.jpg\" alt=\"\" \/><\/span><\/p>\n<p id=\"fs-id1531976\">Notice the change in notation. There are several ways to indicate the determinant, including[latex]\\,\\mathrm{det}\\left(A\\right)\\,[\/latex]and replacing the brackets in a matrix with straight lines,[latex]\\,|A|.[\/latex]<\/p>\n<\/div>\n<div id=\"Example_09_08_01\" class=\"textbox examples\">\n<div id=\"fs-id1046032\">\n<div id=\"fs-id1426119\">\n<h3>Finding the Determinant of a 2 \u00d7 2 Matrix<\/h3>\n<p>Find the determinant of the given matrix.<\/p>\n<div id=\"fs-id1707032\" class=\"unnumbered aligncenter\">[latex]A=\\left[\\begin{array}{cc}5& 2\\\\ -6& 3\\end{array}\\right][\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<div id=\"fs-id1507367\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\mathrm{det}\\left(A\\right)=|\\begin{array}{cc}5& 2\\\\ -6& 3\\end{array}|\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=5\\left(3\\right)-\\left(-6\\right)\\left(2\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=27\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1420496\" class=\"bc-section section\">\n<h3>Using Cramer\u2019s Rule to Solve a System of Two Equations in Two Variables<\/h3>\n<p id=\"fs-id1322833\">We will now introduce a final method for solving systems of equations that uses determinants. Known as <span class=\"no-emphasis\">Cramer\u2019s Rule<\/span>, this technique dates back to the middle of the 18th century and is named for its innovator, the Swiss mathematician Gabriel Cramer (1704-1752), who introduced it in 1750 in Introduction \u00e0 l&#8217;Analyse des lignes Courbes alg\u00e9briques. Cramer\u2019s Rule is a viable and efficient method for finding solutions to systems with an arbitrary number of unknowns, provided that we have the same number of equations as unknowns.<\/p>\n<p id=\"fs-id1531826\">Cramer\u2019s Rule will give us the unique solution to a system of equations, if it exists. However, if the system has no solution or an infinite number of solutions, this will be indicated by a determinant of zero. To find out if the system is inconsistent or dependent, another method, such as elimination, will have to be used.<\/p>\n<p id=\"fs-id1393178\">To understand Cramer\u2019s Rule, let\u2019s look closely at how we solve systems of linear equations using basic row operations. Consider a system of two equations in two variables.<\/p>\n<div id=\"fs-id1279412\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}{a}_{1}x+{b}_{1}y={c}_{1}\\,\\,\\,\\,\\left(1\\right)\\\\ {a}_{2}x+{b}_{2}y={c}_{2}\\,\\,\\,\\,\\left(2\\right)\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1361346\">We eliminate one variable using row operations and solve for the other. Say that we wish to solve for[latex]\\,x.\\,[\/latex]If equation (2) is multiplied by the opposite of the coefficient of[latex]\\,y\\,[\/latex]in equation (1), equation (1) is multiplied by the coefficient of[latex]\\,y\\,[\/latex]in equation (2), and we add the two equations, the variable[latex]\\,y\\,[\/latex]will be eliminated.<\/p>\n<div id=\"fs-id1422407\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\underset{\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_}{\\begin{array}{llll}\\hfill & \\hfill & \\hfill & \\hfill \\\\ \\,\\,\\,\\,{b}_{2}{a}_{1}x+{b}_{2}{b}_{1}y={b}_{2}{c}_{1}\\hfill & \\hfill & \\hfill & \\text{Multiply }{R}_{1}\\text{ by }{b}_{2}\\hfill \\\\ -{b}_{1}{a}_{2}x-{b}_{1}{b}_{2}y=-{b}_{1}{c}_{2}\\hfill & \\hfill & \\hfill & \\text{Multiply }{R}_{2}\\text{ by}-{b}_{1}\\hfill \\end{array}}\\hfill \\\\ \\,\\,\\,\\begin{array}{ll} {b}_{2}{a}_{1}x-{b}_{1}{a}_{2}x={b}_{2}{c}_{1}-{b}_{1}{c}_{2}\\hfill & \\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1370413\">Now, solve for[latex]\\,x.[\/latex]<\/p>\n<div id=\"fs-id1460232\">[latex]\\begin{array}{l}\\,\\,\\,{b}_{2}{a}_{1}x-{b}_{1}{a}_{2}x={b}_{2}{c}_{1}-{b}_{1}{c}_{2}\\hfill \\\\ \\,\\,\\,x\\left({b}_{2}{a}_{1}-{b}_{1}{a}_{2}\\right)={b}_{2}{c}_{1}-{b}_{1}{c}_{2}\\hfill \\\\ \\text{ }x=\\frac{{b}_{2}{c}_{1}-{b}_{1}{c}_{2}}{{b}_{2}{a}_{1}-{b}_{1}{a}_{2}}=\\frac{\\left[\\begin{array}{cc}{c}_{1}& {b}_{1}\\\\ {c}_{2}& {b}_{2}\\end{array}\\right]}{\\left[\\begin{array}{cc}{a}_{1}& {b}_{1}\\\\ {a}_{2}& {b}_{2}\\end{array}\\right]}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1347808\">Similarly, to solve for[latex]\\,y,[\/latex]we will eliminate[latex]\\,x.[\/latex]<\/p>\n<div id=\"fs-id1286733\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\underset{\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_}{\\begin{array}{llll}\\hfill & \\hfill & \\hfill & \\hfill \\\\ \\,\\,\\,\\,{a}_{2}{a}_{1}x+{a}_{2}{b}_{1}y={a}_{2}{c}_{1}\\hfill & \\hfill & \\hfill & \\text{Multiply }{R}_{1}\\text{ by }{a}_{2}\\hfill \\\\ -{a}_{1}{a}_{2}x-{a}_{1}{b}_{2}y=-{a}_{1}{c}_{2}\\hfill & \\hfill & \\hfill & \\text{Multiply }{R}_{2}\\text{ by}-{a}_{1}\\hfill \\end{array}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\begin{array}{ll}{a}_{2}{b}_{1}y-{a}_{1}{b}_{2}y={a}_{2}{c}_{1}-{a}_{1}{c}_{2}\\hfill & \\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1647752\">Solving for[latex]\\,y\\,[\/latex]gives<\/p>\n<div id=\"fs-id1458502\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{a}_{2}{b}_{1}y-{a}_{1}{b}_{2}y={a}_{2}{c}_{1}-{a}_{1}{c}_{2}\\hfill \\\\ y\\left({a}_{2}{b}_{1}-{a}_{1}{b}_{2}\\right)={a}_{2}{c}_{1}-{a}_{1}{c}_{2}\\hfill \\\\ \\text{ }y=\\frac{{a}_{2}{c}_{1}-{a}_{1}{c}_{2}}{{a}_{2}{b}_{1}-{a}_{1}{b}_{2}}=\\frac{{a}_{1}{c}_{2}-{a}_{2}{c}_{1}}{{a}_{1}{b}_{2}-{a}_{2}{b}_{1}}=\\frac{|\\begin{array}{cc}{a}_{1}& {c}_{1}\\\\ {a}_{2}& {c}_{2}\\end{array}|}{|\\begin{array}{cc}{a}_{1}& {b}_{1}\\\\ {a}_{2}& {b}_{2}\\end{array}|}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1281751\">Notice that the denominator for both[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]is the determinant of the coefficient matrix.<\/p>\n<p id=\"fs-id1332312\">We can use these formulas to solve for[latex]\\,x\\,[\/latex]and[latex]\\,y,\\,[\/latex]but Cramer\u2019s Rule also introduces new notation:<\/p>\n<ul id=\"fs-id1279395\">\n<li>[latex]\\,\\,D:[\/latex]determinant of the coefficient matrix<\/li>\n<li>[latex]{D}_{x}:[\/latex]determinant of the numerator in the solution of[latex]x[\/latex]\n<div id=\"fs-id1972131\" class=\"unnumbered aligncenter\">[latex]x=\\frac{{D}_{x}}{D}[\/latex]<\/div>\n<\/li>\n<li>[latex]{D}_{y}:[\/latex]determinant of the numerator in the solution of[latex]\\,y[\/latex]\n<div id=\"fs-id1294030\" class=\"unnumbered aligncenter\">[latex]y=\\frac{{D}_{y}}{D}[\/latex]<\/div>\n<\/li>\n<\/ul>\n<p id=\"fs-id1422623\">The key to Cramer\u2019s Rule is replacing the variable column of interest with the constant column and calculating the determinants. We can then express[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]as a quotient of two determinants.<\/p>\n<div id=\"fs-id1357295\">\n<h3>Cramer\u2019s Rule for 2\u00d72 Systems<\/h3>\n<p id=\"fs-id1664070\">Cramer\u2019s Rule is a method that uses determinants to solve systems of equations that have the same number of equations as variables.<\/p>\n<p id=\"fs-id1660256\">Consider a system of two linear equations in two variables.<\/p>\n<div id=\"fs-id1700631\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}{a}_{1}x+{b}_{1}y={c}_{1}\\\\ {a}_{2}x+{b}_{2}y={c}_{2}\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1338021\">The solution using Cramer\u2019s Rule is given as<\/p>\n<div id=\"fs-id1375892\">[latex]x=\\frac{{D}_{x}}{D}=\\frac{|\\begin{array}{cc}{c}_{1}& {b}_{1}\\\\ {c}_{2}& {b}_{2}\\end{array}|}{|\\begin{array}{cc}{a}_{1}& {b}_{1}\\\\ {a}_{2}& {b}_{2}\\end{array}|},\\,\\,D\\ne 0;\\,\\,\\text{\u200b}\\text{\u200b}\\,y=\\frac{{D}_{y}}{D}=\\frac{|\\begin{array}{cc}{a}_{1}& {c}_{1}\\\\ {a}_{2}& {c}_{2}\\end{array}|}{|\\begin{array}{cc}{a}_{1}& {b}_{1}\\\\ {a}_{2}& {b}_{2}\\end{array}|},\\,\\,D\\ne 0.[\/latex]<\/div>\n<p>If we are solving for[latex]\\,x,\\,[\/latex]the[latex]\\,x\\,[\/latex]column is replaced with the constant column. If we are solving for[latex]\\,y,\\,[\/latex]the[latex]\\,y\\,[\/latex]column is replaced with the constant column.<\/p>\n<\/div>\n<div id=\"Example_09_08_02\" class=\"textbox examples\">\n<div id=\"fs-id1455937\">\n<div id=\"fs-id1455939\">\n<h3>Using Cramer\u2019s Rule to Solve a 2 \u00d7 2 System<\/h3>\n<p id=\"fs-id1421342\">Solve the following[latex]\\,2\\text{ }\u00d7\\text{ }2\\,[\/latex]system using Cramer\u2019s Rule.<\/p>\n<div id=\"fs-id1536728\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}12x+3y=15\\\\ \\text{ }2x-3y=13\\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1459617\">Solve for[latex]\\,x.[\/latex]<\/p>\n<div id=\"fs-id1536461\" class=\"unnumbered aligncenter\">[latex]x=\\frac{{D}_{x}}{D}=\\frac{|\\begin{array}{rr}\\hfill 15& \\hfill 3\\\\ \\hfill 13& \\hfill -3\\end{array}|}{|\\begin{array}{rr}\\hfill 12& \\hfill 3\\\\ \\hfill 2& \\hfill -3\\end{array}|}=\\frac{-45-39}{-36-6}=\\frac{-84}{-42}=2[\/latex]<\/div>\n<p id=\"fs-id1403542\">Solve for[latex]\\,y.[\/latex]<\/p>\n<div id=\"fs-id1697344\" class=\"unnumbered aligncenter\">[latex]y=\\frac{{D}_{y}}{D}=\\frac{|\\begin{array}{rr}\\hfill 12& \\hfill 15\\\\ \\hfill 2& \\hfill 13\\end{array}|}{|\\begin{array}{rr}\\hfill 12& \\hfill 3\\\\ \\hfill 2& \\hfill -3\\end{array}|}=\\frac{156-30}{-36-6}=-\\frac{126}{42}=-3[\/latex]<\/div>\n<p id=\"fs-id1406547\">The solution is[latex]\\,\\left(2,-3\\right).[\/latex]<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1034376\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_09_08_01\">\n<div id=\"fs-id1352281\">\n<p id=\"fs-id1352282\">Use Cramer\u2019s Rule to solve the 2 \u00d7 2 system of equations.<\/p>\n<div id=\"fs-id1647536\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{ }x+2y=-11\\hfill \\\\ -2x+y=-13\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1440640\">[latex]\\left(3,-7\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1594718\" class=\"bc-section section\">\n<h3>Evaluating the Determinant of a 3 \u00d7 3 Matrix<\/h3>\n<p id=\"fs-id1673681\">Finding the determinant of a 2\u00d72 matrix is straightforward, but finding the determinant of a 3\u00d73 matrix is more complicated. One method is to augment the 3\u00d73 matrix with a repetition of the first two columns, giving a 3\u00d75 matrix. Then we calculate the sum of the products of entries <em>down<\/em> each of the three diagonals (upper left to lower right), and subtract the products of entries <em>up<\/em> each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.<\/p>\n<p>Find the <span class=\"no-emphasis\">determinant<\/span> of the 3\u00d73 matrix.<\/p>\n<div id=\"fs-id1505641\" class=\"unnumbered aligncenter\">[latex]A=\\left[\\begin{array}{ccc}{a}_{1}& {b}_{1}& {c}_{1}\\\\ {a}_{2}& {b}_{2}& {c}_{2}\\\\ {a}_{3}& {b}_{3}& {c}_{3}\\end{array}\\right][\/latex]<\/div>\n<ol id=\"fs-id1455932\" type=\"1\">\n<li>Augment[latex]\\,A\\,[\/latex]with the first two columns.\n<div id=\"fs-id1354963\" class=\"unnumbered aligncenter\">[latex]\\mathrm{det}\\left(A\\right)=|\\begin{array}{ccc}{a}_{1}& {b}_{1}& {c}_{1}\\\\ {a}_{2}& {b}_{2}& {c}_{2}\\\\ {a}_{3}& {b}_{3}& {c}_{3}\\end{array}\\,\\,\\,|\\,\\,\\,\\begin{array}{c}{a}_{1}\\\\ {a}_{2}\\\\ {a}_{3}\\end{array}\\,\\,\\,\\,\\begin{array}{c}{b}_{1}\\\\ {b}_{2}\\\\ {b}_{3}\\end{array}|[\/latex]<\/div>\n<\/li>\n<li>From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.<\/li>\n<li>From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.<\/li>\n<\/ol>\n<p><span id=\"fs-id1536830\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154152\/CNX_Precalc_Figure_09_08_002.jpg\" alt=\"\" \/><\/span><\/p>\n<p id=\"fs-id1704961\">The algebra is as follows:<\/p>\n<div id=\"fs-id1704964\" class=\"unnumbered aligncenter\">[latex]|A|={a}_{1}{b}_{2}{c}_{3}+{b}_{1}{c}_{2}{a}_{3}+{c}_{1}{a}_{2}{b}_{3}-{a}_{3}{b}_{2}{c}_{1}-{b}_{3}{c}_{2}{a}_{1}-{c}_{3}{a}_{2}{b}_{1}[\/latex]<\/div>\n<div id=\"Example_09_08_03\" class=\"textbox examples\">\n<div id=\"fs-id1277069\">\n<div id=\"fs-id1277071\">\n<h3>Finding the Determinant of a 3 \u00d7 3 Matrix<\/h3>\n<p id=\"fs-id1529523\">Find the determinant of the 3 \u00d7 3 matrix given<\/p>\n<div id=\"fs-id1529527\" class=\"unnumbered aligncenter\">[latex]A=\\left[\\begin{array}{ccc}0& 2& 1\\\\ 3& -1& 1\\\\ 4& 0& 1\\end{array}\\right][\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>Augment the matrix with the first two columns and then follow the formula. Thus,<\/p>\n<div class=\"unnumbered\">[latex]\\begin{array}{l}|A|=|\\begin{array}{ccc}0& 2& 1\\\\ 3& -1& 1\\\\ 4& 0& 1\\end{array}\\,\\,|\\begin{array}{c}0\\\\ 3\\\\ \\,\\,4\\end{array}\\,\\,\\,\\,\\begin{array}{c}2\\\\ -1\\\\ 0\\end{array}|\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,=0\\left(-1\\right)\\left(1\\right)+2\\left(1\\right)\\left(4\\right)+1\\left(3\\right)\\left(0\\right)-4\\left(-1\\right)\\left(1\\right)-0\\left(1\\right)\\left(0\\right)-1\\left(3\\right)\\left(2\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,=0+8+0+4-0-6\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,=6\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1664049\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_09_08_02\">\n<div id=\"fs-id1608913\">\n<p id=\"fs-id1608914\">Find the determinant of the 3 \u00d7 3 matrix.<\/p>\n<div id=\"fs-id1608917\" class=\"unnumbered aligncenter\">[latex]\\mathrm{det}\\left(A\\right)=|\\begin{array}{ccc}1& -3& 7\\\\ 1& 1& 1\\\\ 1& -2& 3\\end{array}|[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1643971\">[latex]-10[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1424491\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1445990\"><strong>Can we use the same method to find the determinant of a larger matrix?<\/strong><\/p>\n<p id=\"fs-id1445994\"><em>No, this method only works for[latex]\\,2\\text{ }\u00d7\\text{ }2\\,[\/latex]and[latex]\\,\\text{3}\\text{ }\u00d7\\text{ }3\\,[\/latex]matrices. For larger matrices it is best to use a graphing utility or computer software.<\/em><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1420022\" class=\"bc-section section\">\n<h3>Using Cramer\u2019s Rule to Solve a System of Three Equations in Three Variables<\/h3>\n<p id=\"fs-id1433754\">Now that we can find the <span class=\"no-emphasis\">determinant<\/span> of a 3 \u00d7 3 matrix, we can apply <span class=\"no-emphasis\">Cramer\u2019s Rule<\/span> to solve a <span class=\"no-emphasis\">system of three equations in three variables<\/span>. Cramer\u2019s Rule is straightforward, following a pattern consistent with Cramer\u2019s Rule for 2 \u00d7 2 matrices. As the order of the matrix increases to 3 \u00d7 3, however, there are many more calculations required.<\/p>\n<p id=\"fs-id1433166\">When we calculate the determinant to be zero, Cramer\u2019s Rule gives no indication as to whether the system has no solution or an infinite number of solutions. To find out, we have to perform elimination on the system.<\/p>\n<p id=\"fs-id1518566\">Consider a 3 \u00d7 3 system of equations.<\/p>\n<p><span id=\"eip-id1435181\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154154\/9.8.1.jpg\" alt=\"\" \/><\/span><\/p>\n<div id=\"fs-id1452172\" class=\"unnumbered aligncenter\">[latex]x=\\frac{{D}_{x}}{D},y=\\frac{{D}_{y}}{D},z=\\frac{{D}_{z}}{D},D\\ne 0[\/latex]<\/div>\n<p id=\"fs-id1517256\">where<\/p>\n<p><span id=\"eip-id1165135512554\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154155\/9.8.2.jpg\" alt=\"\" \/><\/span><\/p>\n<p id=\"fs-id1702973\">If we are writing the determinant[latex]\\,{D}_{x},[\/latex]we replace the[latex]\\,x\\,[\/latex]column with the constant column. If we are writing the determinant[latex]{D}_{y},[\/latex]we replace the[latex]\\,y\\,[\/latex]column with the constant column. If we are writing the determinant[latex]\\,{D}_{z},[\/latex]we replace the[latex]\\,z\\,[\/latex]column with the constant column. Always check the answer.<\/p>\n<div id=\"Example_09_08_04\" class=\"textbox examples\">\n<div id=\"fs-id1563720\">\n<div id=\"fs-id1182614\">\n<h3>Solving a 3 \u00d7 3 System Using Cramer\u2019s Rule<\/h3>\n<p id=\"fs-id1182620\">Find the solution to the given 3 \u00d7 3 system using Cramer\u2019s Rule.<\/p>\n<div id=\"fs-id1182624\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}x+y-z=6\\\\ 3x-2y+z=-5\\\\ x+3y-2z=14\\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>Use Cramer\u2019s Rule.<\/p>\n<div id=\"fs-id1406667\" class=\"unnumbered aligncenter\">[latex]D=|\\begin{array}{ccc}1& \\,\\,1& -1\\\\ 3& -2& \\,\\,\\,1\\\\ 1& \\,\\,3& -2\\end{array}|,{D}_{x}=|\\begin{array}{ccc}6& 1& -1\\\\ -5& -2& \\,\\,\\,1\\\\ 14& \\,\\,3& -2\\end{array}|,{D}_{y}=|\\begin{array}{ccc}1& \\,6& -1\\\\ 3& -5& \\,\\,1\\\\ 1& 14& -2\\end{array}|,{D}_{z}=|\\begin{array}{ccc}1& \\,1& 6\\\\ 3& -2& -5\\\\ 1& \\,\\,3& 14\\end{array}|[\/latex]<\/div>\n<p id=\"fs-id1436106\">Then,<\/p>\n<div id=\"fs-id1436109\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}x=\\frac{{D}_{x}}{D}=\\frac{-3}{-3}=1\\hfill \\\\ y=\\frac{{D}_{y}}{D}=\\frac{-9}{-3}=3\\hfill \\\\ z=\\frac{{D}_{z}}{D}=\\frac{6}{-3}=-2\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1936367\">The solution is[latex]\\left(1,3,-2\\right).[\/latex]<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1460684\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_09_08_03\">\n<div id=\"fs-id1446687\">\n<p id=\"fs-id1446688\">Use Cramer\u2019s Rule to solve the 3 \u00d7 3 matrix.<\/p>\n<div id=\"fs-id1446691\">[latex]\\begin{array}{r}\\hfill x-3y+7z=13\\\\ \\hfill x+y+z=1\\,\\,\\,\\\\ \\hfill x-2y+3z=4\\,\\,\\,\\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1405450\">[latex]\\left(-2,\\frac{3}{5},\\frac{12}{5}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_09_08_05\" class=\"textbox examples\">\n<div id=\"fs-id1698619\">\n<div id=\"fs-id1698622\">\n<h3>Using Cramer\u2019s Rule to Solve an Inconsistent System<\/h3>\n<p id=\"fs-id1658471\">Solve the system of equations using Cramer\u2019s Rule.<\/p>\n<div id=\"fs-id1658475\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}3x-2y=4\\text{\u2003}\\left(1\\right)\\\\ 6x-4y=0\\text{\u2003}\\left(2\\right)\\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1464455\">We begin by finding the determinants[latex]\\,D,{D}_{x},\\text{and }{D}_{y}.[\/latex]<\/p>\n<div id=\"fs-id1440950\" class=\"unnumbered aligncenter\">[latex]D=|\\begin{array}{cc}3& -2\\\\ 6& -4\\end{array}|=3\\left(-4\\right)-6\\left(-2\\right)=0[\/latex]<\/div>\n<p id=\"fs-id2651884\">We know that a determinant of zero means that either the system has no solution or it has an infinite number of solutions. To see which one, we use the process of elimination. Our goal is to eliminate one of the variables.<\/p>\n<ol id=\"fs-id2651889\" type=\"1\">\n<li>Multiply equation (1) by[latex]\\,-2.[\/latex]<\/li>\n<li>Add the result to equation[latex]\\,\\left(2\\right).[\/latex]<\/li>\n<\/ol>\n<div id=\"fs-id1351077\">[latex]\\begin{array}{l}\\underset{\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_}{\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ -6x+4y\\,\\,\\,\\,=-8\\hfill \\end{array}\\hfill \\\\ \\,\\,\\,6x-4y\\,\\,\\,\\,\\,\\,=\\,\\,\\,\\,0\\hfill \\end{array}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,0\\,\\,\\,\\,\\,\\,=\\,-8\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1352000\">We obtain the equation[latex]\\,0=-8,\\,[\/latex]which is false. Therefore, the system has no solution. Graphing the system reveals two parallel lines. See <a class=\"autogenerated-content\" href=\"#Figure_09_08_003\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_09_08_003\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154202\/CNX_Precalc_Figure_09_08_003.jpg\" alt=\"\" width=\"487\" height=\"441\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_09_08_06\" class=\"textbox examples\">\n<div id=\"fs-id1694152\">\n<div id=\"fs-id1694154\">\n<h3>Use Cramer\u2019s Rule to Solve a Dependent System<\/h3>\n<p id=\"fs-id1694160\">Solve the system with an infinite number of solutions.<\/p>\n<div id=\"fs-id1694163\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{rr}\\hfill x-2y+3z=0& \\hfill \\left(1\\right)\\\\ \\hfill 3x+y-2z=0& \\hfill \\left(2\\right)\\\\ \\hfill 2x-4y+6z=0& \\hfill \\left(3\\right)\\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1360834\">Let\u2019s find the determinant first. Set up a matrix augmented by the first two columns.<\/p>\n<div id=\"fs-id1360839\" class=\"unnumbered aligncenter\">[latex]|\\begin{array}{rrr}\\hfill 1& \\hfill -2& \\hfill 3\\\\ \\hfill 3& \\hfill 1& \\hfill -2\\\\ \\hfill 2& \\hfill -4& \\hfill 6\\end{array}\\text{ }|\\text{ }\\begin{array}{rr}\\hfill 1& \\hfill -2\\\\ \\hfill 3& \\hfill 1\\\\ \\hfill 2& \\hfill -4\\end{array}|[\/latex]<\/div>\n<p id=\"fs-id1523859\">Then,<\/p>\n<div id=\"fs-id1511748\" class=\"unnumbered aligncenter\">[latex]1\\left(1\\right)\\left(6\\right)+\\left(-2\\right)\\left(-2\\right)\\left(2\\right)+3\\left(3\\right)\\left(-4\\right)-2\\left(1\\right)\\left(3\\right)-\\left(-4\\right)\\left(-2\\right)\\left(1\\right)-6\\left(3\\right)\\left(-2\\right)=0[\/latex]<\/div>\n<p id=\"fs-id1699565\">As the determinant equals zero, there is either no solution or an infinite number of solutions. We have to perform elimination to find out.<\/p>\n<ol id=\"fs-id1456188\" type=\"1\">\n<li>Multiply equation (1) by[latex]\\,-2\\,[\/latex]and add the result to equation (3):\n<div id=\"eip-id1165132152839\" class=\"unnumbered\">[latex]\\frac{\\begin{array}{r}\\hfill -2x+4y-6x=0\\\\ \\hfill 2x-4y+6z=0\\end{array}}{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,0=0}[\/latex]<\/div>\n<\/li>\n<li>Obtaining an answer of[latex]\\,0=0,\\,[\/latex]a statement that is always true, means that the system has an infinite number of solutions. Graphing the system, we can see that two of the planes are the same and they both intersect the third plane on a line. See <a class=\"autogenerated-content\" href=\"#Figure_09_08_005\">(Figure).<\/a><br \/>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154205\/CNX_Precalc_Figure_09_08_005.jpg\" alt=\"\" width=\"487\" height=\"214\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/figcaption><\/figure>\n<\/details>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1651973\" class=\"bc-section section\">\n<h3>Understanding Properties of Determinants<\/h3>\n<p id=\"fs-id1651978\">There are many <span class=\"no-emphasis\">properties of determinants<\/span>. Listed here are some properties that may be helpful in calculating the determinant of a matrix.<\/p>\n<div id=\"fs-id1651986\">\n<h3>Properties of Determinants<\/h3>\n<ol id=\"fs-id1459457\" type=\"1\">\n<li>If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.<\/li>\n<li>When two rows are interchanged, the determinant changes sign.<\/li>\n<li>If either two rows or two columns are identical, the determinant equals zero.<\/li>\n<li>If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.<\/li>\n<li>The determinant of an inverse matrix[latex]\\,{A}^{-1}\\,[\/latex]is the reciprocal of the determinant of the matrix[latex]\\,A.[\/latex]<\/li>\n<li>If any row or column is multiplied by a constant, the determinant is multiplied by the same factor.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_09_08_07\" class=\"textbox examples\">\n<div id=\"fs-id1531866\">\n<div id=\"fs-id1531868\">\n<h3>Illustrating Properties of Determinants<\/h3>\n<p id=\"fs-id1531874\">Illustrate each of the properties of determinants.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1531880\">Property 1 states that if the matrix is in upper triangular form, the determinant is the product of the entries down the main diagonal.<\/p>\n<div id=\"fs-id1685740\" class=\"unnumbered aligncenter\">[latex]A=\\left[\\begin{array}{rrr}\\hfill 1& \\hfill \\,\\,2& \\hfill 3\\\\ \\hfill 0& \\hfill \\,\\,2& \\hfill 1\\\\ \\hfill 0& \\hfill \\,\\,0& \\hfill -1\\end{array}\\right][\/latex]<\/div>\n<p id=\"fs-id1370716\">Augment[latex]\\,A\\,[\/latex]with the first two columns.<\/p>\n<div id=\"fs-id1504903\" class=\"unnumbered aligncenter\">[latex]A=\\left[\\begin{array}{ccc}1& 2& 3\\\\ 0& 2& 1\\\\ 0& 0& -1\\end{array}|\\,\\,\\,\\begin{array}{c}1\\\\ 0\\\\ 0\\end{array}\\,\\,\\,\\,\\begin{array}{c}2\\\\ 2\\\\ 0\\end{array}\\right][\/latex]<\/div>\n<p id=\"fs-id1608365\">Then<\/p>\n<div id=\"fs-id1608368\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\mathrm{det}\\left(A\\right)=1\\left(2\\right)\\left(-1\\right)+2\\left(1\\right)\\left(0\\right)+3\\left(0\\right)\\left(0\\right)-0\\left(2\\right)\\left(3\\right)-0\\left(1\\right)\\left(1\\right)+1\\left(0\\right)\\left(2\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=-2\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1429354\">Property 2 states that interchanging rows changes the sign. Given<\/p>\n<div class=\"unnumbered\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ A=\\left[\\begin{array}{cc}-1& 5\\\\ 4& -3\\end{array}\\right],\\,\\,\\mathrm{det}\\left(A\\right)=\\left(-1\\right)\\left(-3\\right)-\\left(4\\right)\\left(5\\right)=3-20=-17\\end{array}\\hfill \\\\ \\hfill \\\\ B=\\left[\\begin{array}{cc}4& -3\\\\ -1& 5\\end{array}\\right],\\,\\,\\mathrm{det}\\left(B\\right)=\\left(4\\right)\\left(5\\right)-\\left(-1\\right)\\left(-3\\right)=20-3=17\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1433736\">Property 3 states that if two rows or two columns are identical, the determinant equals zero.<\/p>\n<div class=\"unnumbered\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,A=\\left[\\begin{array}{ccc}1& 2& 2\\\\ 2& 2& 2\\\\ -1& 2& 2\\end{array}\\text{ }|\\text{ }\\begin{array}{c}1\\\\ 2\\\\ -1\\end{array} \\begin{array}{c}2\\\\ 2\\\\ 2\\end{array}\\right]\\hfill \\\\ \\hfill \\\\ \\mathrm{det}\\left(A\\right)=1\\left(2\\right)\\left(2\\right)+2\\left(2\\right)\\left(-1\\right)+2\\left(2\\right)\\left(2\\right)+1\\left(2\\right)\\left(2\\right)-2\\left(2\\right)\\left(1\\right)-2\\left(2\\right)\\left(2\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=4-4+8+4-4-8=0\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1374562\">Property 4 states that if a row or column equals zero, the determinant equals zero. Thus,<\/p>\n<div id=\"fs-id1408472\" class=\"unnumbered aligncenter\">[latex]A=\\left[\\begin{array}{cc}1& 2\\\\ 0& 0\\end{array}\\right],\\,\\,\\,\\mathrm{det}\\left(A\\right)=1\\left(0\\right)-2\\left(0\\right)=0[\/latex]<\/div>\n<p id=\"fs-id1434974\">Property 5 states that the determinant of an inverse matrix[latex]\\,{A}^{-1}\\,[\/latex]is the reciprocal of the determinant[latex]\\,A.\\,[\/latex]Thus,<\/p>\n<div id=\"fs-id1371025\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,A=\\left[\\begin{array}{cc}1& 2\\\\ 3& 4\\end{array}\\right],\\mathrm{det}\\left(A\\right)=1\\left(4\\right)-3\\left(2\\right)=-2\\hfill \\\\ \\hfill \\\\ {A}^{-1}=\\left[\\begin{array}{cc}-2& 1\\\\ \\frac{3}{2}& -\\frac{1}{2}\\end{array}\\right],\\mathrm{det}\\left({A}^{-1}\\right)=-2\\left(-\\frac{1}{2}\\right)-\\left(\\frac{3}{2}\\right)\\left(1\\right)=-\\frac{1}{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1455730\">Property 6 states that if any row or column of a matrix is multiplied by a constant, the determinant is multiplied by the same factor. Thus,<\/p>\n<div id=\"fs-id1455735\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}A=\\left[\\begin{array}{cc}1& 2\\\\ 3& 4\\end{array}\\right],\\mathrm{det}\\left(A\\right)=1\\left(4\\right)-2\\left(3\\right)=-2\\hfill \\\\ \\hfill \\\\ B=\\left[\\begin{array}{cc}2\\left(1\\right)& 2\\left(2\\right)\\\\ 3& 4\\end{array}\\right],\\mathrm{det}\\left(B\\right)=2\\left(4\\right)-3\\left(4\\right)=-4\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_09_08_08\" class=\"textbox examples\">\n<div id=\"fs-id1417083\">\n<div id=\"fs-id1417085\">\n<h3>Using Cramer\u2019s Rule and Determinant Properties to Solve a System<\/h3>\n<p id=\"fs-id1417091\">Find the solution to the given 3 \u00d7 3 system.<\/p>\n<div id=\"fs-id1417094\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}2x+4y+4z=2\\hfill & \\left(1\\right)\\hfill \\\\ 3x+7y+7z=-5\\hfill & \\left(2\\right)\\hfill \\\\ \\text{ }x+2y+2z=4\\hfill & \\left(3\\right)\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1653554\">Using <span class=\"no-emphasis\">Cramer\u2019s Rule<\/span>, we have<\/p>\n<div id=\"fs-id1407186\" class=\"unnumbered aligncenter\">[latex]D=|\\begin{array}{ccc}2& 4& 4\\\\ 3& 7& 7\\\\ 1& 2& 2\\end{array}|[\/latex]<\/div>\n<p id=\"fs-id1676090\">Notice that the second and third columns are identical. According to Property 3, the determinant will be zero, so there is either no solution or an infinite number of solutions. We have to perform elimination to find out.<\/p>\n<ol id=\"fs-id1676095\" type=\"1\">\n<li>Multiply equation (3) by \u20132 and add the result to equation (1).\n<div id=\"fs-id1407844\" class=\"unnumbered aligncenter\">[latex]\\frac{\\begin{array}{l}-2x-4y-4x=-8\\hfill \\\\ \\text{ }2x+4y+4z=2\\,\\,\\,\\,\\,\\hfill \\end{array}}{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,0=-6}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p id=\"fs-id1422312\">Obtaining a statement that is a contradiction means that the system has no solution.<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1422319\" class=\"precalculus media\">\n<p id=\"fs-id1677169\">Access these online resources for additional instruction and practice with Cramer\u2019s Rule.<\/p>\n<ul id=\"fs-id1677174\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/system2cramer\">Solve a System of Two Equations Using Cramer&#8217;s Rule<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/system3cramer\">Solve a Systems of Three Equations using Cramer&#8217;s Rule<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1677186\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1600273\">\n<li>The determinant for[latex]\\,\\left[\\begin{array}{cc}a& b\\\\ c& d\\end{array}\\right]\\,[\/latex]is[latex]\\,ad-bc.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_09_08_01\">(Figure)<\/a>.<\/li>\n<li>Cramer\u2019s Rule replaces a variable column with the constant column. Solutions are[latex]\\,x=\\frac{{D}_{x}}{D},y=\\frac{{D}_{y}}{D}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_09_08_02\">(Figure)<\/a>.<\/li>\n<li>To find the determinant of a 3\u00d73 matrix, augment with the first two columns. Add the three diagonal entries (upper left to lower right) and subtract the three diagonal entries (lower left to upper right). See <a class=\"autogenerated-content\" href=\"#Example_09_08_03\">(Figure)<\/a>.<\/li>\n<li>To solve a system of three equations in three variables using Cramer\u2019s Rule, replace a variable column with the constant column for each desired solution:[latex]\\,x=\\frac{{D}_{x}}{D},y=\\frac{{D}_{y}}{D},z=\\frac{{D}_{z}}{D}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_09_08_04\">(Figure)<\/a>.<\/li>\n<li>Cramer\u2019s Rule is also useful for finding the solution of a system of equations with no solution or infinite solutions. See <a class=\"autogenerated-content\" href=\"#Example_09_08_05\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_09_08_06\">(Figure)<\/a>.<\/li>\n<li>Certain properties of determinants are useful for solving problems. For example:\n<ul id=\"eip-id1165135344724\">\n<li>If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.<\/li>\n<li>When two rows are interchanged, the determinant changes sign.<\/li>\n<li>If either two rows or two columns are identical, the determinant equals zero.<\/li>\n<li>If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.<\/li>\n<li>The determinant of an inverse matrix[latex]\\,{A}^{-1}\\,[\/latex]is the reciprocal of the determinant of the matrix[latex]\\,A.[\/latex]<\/li>\n<li>If any row or column is multiplied by a constant, the determinant is multiplied by the same factor. See <a class=\"autogenerated-content\" href=\"#Example_09_08_07\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_09_08_08\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1408613\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1408620\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1405255\">\n<div id=\"fs-id1405256\">\n<p id=\"fs-id1405257\">Explain why we can always evaluate the determinant of a square matrix.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>A determinant is the sum and products of the entries in the matrix, so you can always evaluate that product\u2014even if it does end up being 0.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1405269\">\n<div id=\"fs-id1405270\">\n<p id=\"fs-id1405271\">Examining Cramer\u2019s Rule, explain why there is no unique solution to the system when the determinant of your matrix is 0. For simplicity, use a[latex]\\,2\\,\u00d7\\,2\\,[\/latex]matrix.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1394594\">\n<div id=\"fs-id1394595\">\n<p id=\"fs-id1394596\">Explain what it means in terms of an inverse for a matrix to have a 0 determinant.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1394601\">The inverse does not exist.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1394606\">\n<div id=\"fs-id1394607\">\n<p id=\"fs-id1394608\">The determinant of[latex]\\,2\\,\u00d7\\,2\\,[\/latex]matrix[latex]\\,A\\,[\/latex]is 3. If you switch the rows and multiply the first row by 6 and the second row by 2, explain how to find the determinant and provide the answer.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1417224\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1417229\">For the following exercises, find the determinant.<\/p>\n<div id=\"fs-id1417233\">\n<div id=\"fs-id1417234\">\n<p id=\"fs-id1394308\">[latex]|\\begin{array}{cc}1& 2\\\\ 3& 4\\end{array}|[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1394170\">[latex]-2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1522034\">\n<div id=\"fs-id1522035\">\n<p id=\"fs-id1522036\">[latex]|\\begin{array}{rr}\\hfill -1& \\hfill 2\\\\ \\hfill 3& \\hfill -4\\end{array}|[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1434436\">\n<div id=\"fs-id1434437\">\n<p id=\"fs-id1434438\">[latex]|\\begin{array}{rr}\\hfill 2& \\hfill -5\\\\ \\hfill -1& \\hfill 6\\end{array}|[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1430900\">[latex]7[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1430911\">\n<div id=\"fs-id1430912\">\n<p id=\"fs-id1430913\">[latex]|\\begin{array}{cc}-8& 4\\\\ -1& 5\\end{array}|[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1677911\">\n<div id=\"fs-id1677912\">\n<p id=\"fs-id1677913\">[latex]|\\begin{array}{rr}\\hfill 1& \\hfill 0\\\\ \\hfill 3& \\hfill -4\\end{array}|[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1395220\">[latex]-4[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1408793\">\n<div id=\"fs-id1408794\">\n<p id=\"fs-id1408795\">[latex]|\\begin{array}{rr}\\hfill 10& \\hfill 20\\\\ \\hfill 0& \\hfill -10\\end{array}|[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1407150\">\n<div id=\"fs-id1407151\">\n<p id=\"fs-id1407152\">[latex]|\\begin{array}{cc}10& 0.2\\\\ 5& 0.1\\end{array}|[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1660075\">[latex]0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1660084\">\n<div id=\"fs-id1660085\">\n<p id=\"fs-id1660086\">[latex]|\\begin{array}{rr}\\hfill 6& \\hfill -3\\\\ \\hfill 8& \\hfill 4\\end{array}|[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1587739\">\n<div id=\"fs-id1587740\">\n<p id=\"fs-id1587741\">[latex]|\\begin{array}{rr}\\hfill -2& \\hfill -3\\\\ \\hfill 3.1& \\hfill 4,000\\end{array}|[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id885813\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id885815\">[latex]-7,990.7[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div>[latex]|\\begin{array}{rr}\\hfill -1.1& \\hfill 0.6\\\\ \\hfill 7.2& \\hfill -0.5\\end{array}|[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1513986\">\n<div id=\"fs-id1513987\">\n<p id=\"fs-id1513988\">[latex]|\\begin{array}{rrr}\\hfill -1& \\hfill 0& \\hfill 0\\\\ \\hfill 0& \\hfill 1& \\hfill 0\\\\ \\hfill 0& \\hfill 0& \\hfill -3\\end{array}|[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1339464\">[latex]3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1339474\">\n<div id=\"fs-id1339475\">\n<p id=\"fs-id1339476\">[latex]|\\begin{array}{rrr}\\hfill -1& \\hfill 4& \\hfill 0\\\\ \\hfill 0& \\hfill 2& \\hfill 3\\\\ \\hfill 0& \\hfill 0& \\hfill -3\\end{array}|[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1586650\">\n<div id=\"fs-id1586652\">\n<p id=\"fs-id1586653\">[latex]|\\begin{array}{ccc}1& 0& 1\\\\ 0& 1& 0\\\\ 1& 0& 0\\end{array}|[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1601846\">[latex]-1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id881216\">\n<div id=\"fs-id881217\">\n<p id=\"fs-id881218\">[latex]|\\begin{array}{rrr}\\hfill 2& \\hfill -3& \\hfill 1\\\\ \\hfill 3& \\hfill -4& \\hfill 1\\\\ \\hfill -5& \\hfill 6& \\hfill 1\\end{array}|[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2692384\">\n<div id=\"fs-id2692386\">\n<p id=\"fs-id2692387\">[latex]|\\begin{array}{rrr}\\hfill -2& \\hfill 1& \\hfill 4\\\\ \\hfill -4& \\hfill 2& \\hfill -8\\\\ \\hfill 2& \\hfill -8& \\hfill -3\\end{array}|[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1663783\">[latex]224[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1663796\">\n<div id=\"fs-id1663797\">\n<p id=\"fs-id1663798\">[latex]|\\begin{array}{rrr}\\hfill 6& \\hfill -1& \\hfill 2\\\\ \\hfill -4& \\hfill -3& \\hfill 5\\\\ \\hfill 1& \\hfill 9& \\hfill -1\\end{array}|[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1365782\">\n<div id=\"fs-id1365783\">\n<p id=\"fs-id1365784\">[latex]|\\begin{array}{rrr}\\hfill 5& \\hfill 1& \\hfill -1\\\\ \\hfill 2& \\hfill 3& \\hfill 1\\\\ \\hfill 3& \\hfill -6& \\hfill -3\\end{array}|[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1616218\">[latex]15[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1616230\">\n<div id=\"fs-id1616232\">\n<p id=\"fs-id1616233\">[latex]|\\begin{array}{rrr}\\hfill 1.1& \\hfill 2& \\hfill -1\\\\ \\hfill -4& \\hfill 0& \\hfill 0\\\\ \\hfill 4.1& \\hfill -0.4& \\hfill 2.5\\end{array}|[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1370226\">\n<div id=\"fs-id1370227\">\n<p id=\"fs-id1370228\">[latex]|\\begin{array}{rrr}\\hfill 2& \\hfill -1.6& \\hfill 3.1\\\\ \\hfill 1.1& \\hfill 3& \\hfill -8\\\\ \\hfill -9.3& \\hfill 0& \\hfill 2\\end{array}|[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1534877\">[latex]-17.03[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1534892\">\n<div id=\"fs-id1534893\">\n<p id=\"fs-id1534894\">[latex]|\\begin{array}{ccc}-\\frac{1}{2}& \\frac{1}{3}& \\frac{1}{4}\\\\ \\frac{1}{5}& -\\frac{1}{6}& \\frac{1}{7}\\\\ 0& 0& \\frac{1}{8}\\end{array}|[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1891694\">For the following exercises, solve the system of linear equations using Cramer\u2019s Rule.<\/p>\n<div id=\"fs-id1891699\">\n<div id=\"fs-id1891700\">\n<p id=\"fs-id1891701\">[latex]\\begin{array}{l}2x-3y=-1\\\\ 4x+5y=9\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1694937\">[latex]\\left(1,1\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1914122\">\n<div id=\"fs-id1914123\">\n<p id=\"fs-id1914124\">[latex]\\begin{array}{r}5x-4y=2\\\\ -4x+7y=6\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1704920\">\n<div id=\"fs-id1704921\">\n<p id=\"fs-id1704922\">[latex]\\begin{array}{l}\\text{ }6x-3y=2\\,\\,\\,\\,\\,\\hfill \\\\ -8x+9y=-1\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1433846\">[latex]\\left(\\frac{1}{2},\\frac{1}{3}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1381750\">\n<div id=\"fs-id1675292\">\n<p id=\"fs-id1675294\">[latex]\\begin{array}{l}2x+6y=12\\\\ 5x-2y=13\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1408543\">\n<div id=\"fs-id1408544\">\n<p id=\"fs-id1408546\">[latex]\\begin{array}{l}4x+3y=23\\,\\,\\hfill \\\\ \\text{ }2x-y=-1\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1516926\">[latex]\\left(2,5\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1513467\">\n<div id=\"fs-id1513468\">\n<p id=\"fs-id1513469\">[latex]\\begin{array}{l}10x-6y=2\\,\\,\\,\\,\\hfill \\\\ -5x+8y=-1\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1512009\">\n<div id=\"fs-id1512010\">\n<p id=\"fs-id1512012\">[latex]\\begin{array}{l}4x-3y=-3\\\\ 2x+6y=-4\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1615360\">[latex]\\left(-1,-\\frac{1}{3}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1523200\">\n<div id=\"fs-id1514396\">\n<p id=\"fs-id1514397\">[latex]\\begin{array}{r}4x-5y=7\\\\ -3x+9y=0\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1701646\">\n<div id=\"fs-id1701647\">\n<p id=\"fs-id1701648\">[latex]\\begin{array}{l}4x+10y=180\\,\\,\\,\\,\\hfill \\\\ -3x-5y=-105\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1530071\">[latex]\\left(15,12\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1394699\">\n<div id=\"fs-id1394700\">\n<p id=\"fs-id1394702\">[latex]\\begin{array}{l}\\text{ }8x-2y=-3\\hfill \\\\ -4x+6y=4\\,\\,\\,\\,\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1517129\">For the following exercises, solve the system of linear equations using Cramer\u2019s Rule.<\/p>\n<div id=\"fs-id1517134\">\n<div id=\"fs-id1517135\">[latex]\\begin{array}{l}\\text{ }x+2y-4z=-1\\hfill \\\\ \\text{ }7x+3y+5z=26\\,\\,\\hfill \\\\ -2x-6y+7z=-6\\hfill \\end{array}[\/latex]<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1530969\">[latex]\\left(1,3,2\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1699396\">\n<div id=\"fs-id1699397\">\n<p id=\"fs-id1699398\">[latex]\\begin{array}{l}-5x+2y-4z=-47\\hfill \\\\ \\text{ }4x-3y-z=-94\\hfill \\\\ \\text{ }3x-3y+2z=94\\,\\,\\,\\,\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1351513\">\n<div id=\"fs-id1351514\">\n<p id=\"fs-id1351515\">[latex]\\begin{array}{l}\\text{ }4x+5y-z=-7\\hfill \\\\ -2x-9y+2z=8\\,\\,\\,\\,\\hfill \\\\ \\text{ }5y+7z=21\\,\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1699357\">[latex]\\left(-1,0,3\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1355045\">\n<div id=\"fs-id1355046\">\n<p id=\"fs-id1355047\">[latex]\\begin{array}{r}4x-3y+4z=10\\\\ 5x-2z=-2\\\\ 3x+2y-5z=-9\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1410018\">\n<div id=\"fs-id1410019\">\n<p id=\"fs-id1410020\">[latex]\\begin{array}{l}4x-2y+3z=6\\,\\,\\,\\hfill \\\\ \\text{ }-6x+y=-2\\hfill \\\\ 2x+7y+8z=24\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1506671\">[latex]\\left(\\frac{1}{2},1,2\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1430552\">\n<div id=\"fs-id1430553\">\n<p id=\"fs-id1430554\">[latex]\\begin{array}{r}\\hfill 5x+2y-z=1\\,\\,\\,\\,\\,\\\\ \\hfill -7x-8y+3z=1.5\\\\ \\hfill 6x-12y+z=7\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1348575\">\n<div id=\"fs-id1348576\">\n<p id=\"fs-id1348577\">[latex]\\begin{array}{l}\\text{ }13x-17y+16z=73\\,\\,\\,\\,\\hfill \\\\ -11x+15y+17z=61\\,\\,\\,\\,\\hfill \\\\ \\text{ }46x+10y-30z=-18\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]\\left(2,1,4\\right)[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1519001\">\n<div id=\"fs-id1519002\">\n<p id=\"fs-id1519003\">[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ -4x-3y-8z=-7\\hfill \\end{array}\\hfill \\\\ \\text{ }2x-9y+5z=0.5\\hfill \\\\ \\text{ }5x-6y-5z=-2\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1601296\">\n<div id=\"fs-id1601297\">\n<p id=\"fs-id1601298\">[latex]\\begin{array}{l}\\text{ }4x-6y+8z=10\\,\\,\\hfill \\\\ -2x+3y-4z=-5\\hfill \\\\ \\text{ }x+y+z=1\\,\\,\\,\\,\\,\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1673950\">Infinite solutions<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1446784\">\n<div id=\"fs-id1446785\">\n<p id=\"fs-id1446786\">[latex]\\begin{array}{r}\\hfill 4x-6y+8z=10\\,\\,\\,\\,\\,\\\\ \\hfill -2x+3y-4z=-5\\,\\,\\,\\\\ \\hfill 12x+18y-24z=-30\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1487525\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1487530\">For the following exercises, use the determinant function on a graphing utility.<\/p>\n<div id=\"fs-id1487533\">\n<div id=\"fs-id1487534\">[latex]|\\begin{array}{rrrr}\\hfill 1& \\hfill 0& \\hfill 8& \\hfill 9\\\\ \\hfill 0& \\hfill 2& \\hfill 1& \\hfill 0\\\\ \\hfill 1& \\hfill 0& \\hfill 3& \\hfill 0\\\\ \\hfill 0& \\hfill 2& \\hfill 4& \\hfill 3\\end{array}|[\/latex]<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1597711\">[latex]24[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1597725\">\n<div id=\"fs-id1597726\">\n<p id=\"fs-id1597727\">[latex]|\\begin{array}{rrrr}\\hfill 1& \\hfill 0& \\hfill 2& \\hfill 1\\\\ \\hfill 0& \\hfill -9& \\hfill 1& \\hfill 3\\\\ \\hfill 3& \\hfill 0& \\hfill -2& \\hfill -1\\\\ \\hfill 0& \\hfill 1& \\hfill 1& \\hfill -2\\end{array}|[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1700780\">\n<div id=\"fs-id1700781\">\n<p id=\"fs-id1700782\">[latex]|\\begin{array}{rrrr}\\hfill \\frac{1}{2}& \\hfill 1& \\hfill 7& \\hfill 4\\\\ \\hfill 0& \\hfill \\frac{1}{2}& \\hfill 100& \\hfill 5\\\\ \\hfill 0& \\hfill 0& \\hfill 2& \\hfill 2,000\\\\ \\hfill 0& \\hfill 0& \\hfill 0& \\hfill 2\\end{array}|[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1905255\">[latex]1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1905264\">\n<div id=\"fs-id1905265\">\n<p id=\"fs-id1905266\">[latex]|\\begin{array}{rrrr}\\hfill 1& \\hfill 0& \\hfill 0& \\hfill 0\\\\ \\hfill 2& \\hfill 3& \\hfill 0& \\hfill 0\\\\ \\hfill 4& \\hfill 5& \\hfill 6& \\hfill 0\\\\ \\hfill 7& \\hfill 8& \\hfill 9& \\hfill 0\\end{array}|[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1420602\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<p id=\"fs-id1702572\">For the following exercises, create a system of linear equations to describe the behavior. Then, calculate the determinant. Will there be a unique solution? If so, find the unique solution.<\/p>\n<div id=\"fs-id1702577\">\n<div id=\"fs-id1702578\">\n<p id=\"fs-id1702579\">Two numbers add up to 56. One number is 20 less than the other.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1702584\">Yes; 18, 38<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1702588\">\n<div id=\"fs-id1702590\">\n<p id=\"fs-id1702591\">Two numbers add up to 104. If you add two times the first number plus two times the second number, your total is 208<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1702595\">\n<div id=\"fs-id1791156\">\n<p id=\"fs-id1791157\">Three numbers add up to 106. The first number is 3 less than the second number. The third number is 4 more than the first number.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1791164\">Yes; 33, 36, 37<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1791168\">\n<div id=\"fs-id1791169\">\n<p id=\"fs-id1791170\">Three numbers add to 216. The sum of the first two numbers is 112. The third number is 8 less than the first two numbers combined.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1791174\">For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer\u2019s Rule.<\/p>\n<div id=\"fs-id1791180\">\n<div id=\"fs-id1791181\">\n<p id=\"fs-id1791182\">You invest $10,000 into two accounts, which receive 8% interest and 5% interest. At the end of a year, you had $10,710 in your combined accounts. How much was invested in each account?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1354932\">$7,000 in first account, $3,000 in second account.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1354936\">\n<div id=\"fs-id1354937\">\n<p id=\"fs-id1354938\">You invest $80,000 into two accounts, $22,000 in one account, and $58,000 in the other account. At the end of one year, assuming simple interest, you have earned $2,470 in interest. The second account receives half a percent less than twice the interest on the first account. What are the interest rates for your accounts?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1354944\">\n<div id=\"fs-id1354946\">\n<p id=\"fs-id1354947\">A movie theater needs to know how many adult tickets and children tickets were sold out of the 1,200 total tickets. If children\u2019s tickets are $5.95, adult tickets are $11.15, and the total amount of revenue was $12,756, how many children\u2019s tickets and adult tickets were sold?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1584198\">120 children, 1,080 adult<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1584202\">\n<div id=\"fs-id1584203\">\n<p id=\"fs-id1584204\">A concert venue sells single tickets for $40 each and couple\u2019s tickets for $65. If the total revenue was $18,090 and the 321 tickets were sold, how many single tickets and how many couple\u2019s tickets were sold?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1584211\">\n<div id=\"fs-id1584212\">\n<p id=\"fs-id1584213\">You decide to paint your kitchen green. You create the color of paint by mixing yellow and blue paints. You cannot remember how many gallons of each color went into your mix, but you know there were 10 gal total. Additionally, you kept your receipt, and know the total amount spent was $29.50. If each gallon of yellow costs $2.59, and each gallon of blue costs $3.19, how many gallons of each color go into your green mix?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1584221\">4 gal yellow, 6 gal blue<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1698980\">\n<div id=\"fs-id1698981\">\n<p id=\"fs-id1698982\">You sold two types of scarves at a farmers\u2019 market and would like to know which one was more popular. The total number of scarves sold was 56, the yellow scarf cost $10, and the purple scarf cost $11. If you had total revenue of $583, how many yellow scarves and how many purple scarves were sold?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1698991\">\n<div id=\"fs-id1698992\">\n<p id=\"fs-id1698993\">Your garden produced two types of tomatoes, one green and one red. The red weigh 10 oz, and the green weigh 4 oz. You have 30 tomatoes, and a total weight of 13 lb, 14 oz. How many of each type of tomato do you have?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1699000\">13 green tomatoes, 17 red tomatoes<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1648791\">\n<div id=\"fs-id1648792\">\n<p id=\"fs-id1648793\">At a market, the three most popular vegetables make up 53% of vegetable sales. Corn has 4% higher sales than broccoli, which has 5% more sales than onions. What percentage does each vegetable have in the market share?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1648798\">\n<div id=\"fs-id1648799\">\n<p id=\"fs-id1648800\">At the same market, the three most popular fruits make up 37% of the total fruit sold. Strawberries sell twice as much as oranges, and kiwis sell one more percentage point than oranges. For each fruit, find the percentage of total fruit sold.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1648808\">Strawberries 18%, oranges 9%, kiwi 10%<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1648813\">\n<div id=\"fs-id1648814\">\n<p id=\"fs-id1648815\">Three bands performed at a concert venue. The first band charged $15 per ticket, the second band charged $45 per ticket, and the final band charged $22 per ticket. There were 510 tickets sold, for a total of $12,700. If the first band had 40 more audience members than the second band, how many tickets were sold for each band?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1637276\">\n<div id=\"fs-id1637277\">\n<p id=\"fs-id1637278\">A movie theatre sold tickets to three movies. The tickets to the first movie were $5, the tickets to the second movie were $11, and the third movie was $12. 100 tickets were sold to the first movie. The total number of tickets sold was 642, for a total revenue of $6,774. How many tickets for each movie were sold?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1637286\">100 for movie 1, 230 for movie 2, 312 for movie 3<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1637290\">\n<div id=\"fs-id1637291\">\n<p id=\"fs-id1637292\">Men aged 20\u201329, 30\u201339, and 40\u201349 made up 78% of the population at a prison last year. This year, the same age groups made up 82.08% of the population. The 20\u201329 age group increased by 20%, the 30\u201339 age group increased by 2%, and the 40\u201349 age group decreased to[latex]\\,\\frac{3}{4}\\,[\/latex]of their previous population. Originally, the 30\u201339 age group had 2% more prisoners than the 20\u201329 age group. Determine the prison population percentage for each age group last year.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1425381\">\n<div id=\"fs-id1425382\">\n<p id=\"fs-id1425383\">At a women\u2019s prison down the road, the total number of inmates aged 20\u201349 totaled 5,525. This year, the 20\u201329 age group increased by 10%, the 30\u201339 age group decreased by 20%, and the 40\u201349 age group doubled. There are now 6,040 prisoners. Originally, there were 500 more in the 30\u201339 age group than the 20\u201329 age group. Determine the prison population for each age group last year.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1425394\">20\u201329: 2,100, 30\u201339: 2,600, 40\u201349: 825<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1425398\">For the following exercises, use this scenario: A health-conscious company decides to make a trail mix out of almonds, dried cranberries, and chocolate-covered cashews. The nutritional information for these items is shown in <a class=\"autogenerated-content\" href=\"#Table_09_08_01\">(Figure)<\/a>.<\/p>\n<table id=\"Table_09_08_01\" summary=\"..\">\n<thead>\n<tr>\n<th><\/th>\n<th>Fat (g)<\/th>\n<th>Protein (g)<\/th>\n<th>Carbohydrates (g)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><strong>Almonds (10)<\/strong><\/td>\n<td>6<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td><strong>Cranberries (10)<\/strong><\/td>\n<td>0.02<\/td>\n<td>0<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td><strong>Cashews (10)<\/strong><\/td>\n<td>7<\/td>\n<td>3.5<\/td>\n<td>5.5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1394923\">\n<div id=\"fs-id1394924\">\n<p id=\"fs-id1394925\">For the special \u201clow-carb\u201dtrail mix, there are 1,000 pieces of mix. The total number of carbohydrates is 425 g, and the total amount of fat is 570.2 g. If there are 200 more pieces of cashews than cranberries, how many of each item is in the trail mix?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1394933\">\n<div id=\"fs-id1394934\">\n<p id=\"fs-id1394935\">For the \u201chiking\u201d mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. If there is the same amount of almonds as cashews, how many of each item is in the trail mix?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1647224\">300 almonds, 400 cranberries, 300 cashews<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1647228\">\n<div id=\"fs-id1647230\">\n<p id=\"fs-id1647231\">For the \u201cenergy-booster\u201d mix, there are 1,000 pieces in the mix, containing 145 g of protein and 625 g of carbohydrates. If the number of almonds and cashews summed together is equivalent to the amount of cranberries, how many of each item is in the trail mix?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"review-exercises textbox exercises\">\n<h3>Review Exercises<\/h3>\n<div id=\"fs-id2876816\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/0bb1dacf-671b-4fe5-a5a4-e67ad9a48d54\">Systems of Linear Equations: Two Variables<\/a><\/h4>\n<p id=\"fs-id1460571\">For the following exercises, determine whether the ordered pair is a solution to the system of equations.<\/p>\n<div id=\"fs-id1460575\">\n<div id=\"fs-id1460576\">\n<p id=\"fs-id1460577\">[latex]\\begin{array}{l}3x-y=4\\\\ x+4y=-3\\,\\end{array}[\/latex]and[latex]\\,\\left(-1,1\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1395131\">No<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1395135\">\n<div>\n<p id=\"fs-id1395137\">[latex]\\begin{array}{l}6x-2y=24\\\\ -3x+3y=18\\,\\end{array}[\/latex]and[latex]\\,\\left(9,15\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1519982\">For the following exercises, use substitution to solve the system of equations.<\/p>\n<div id=\"fs-id1519985\">\n<div id=\"fs-id1519986\">\n<p id=\"fs-id1519987\">[latex]\\begin{array}{l}10x+5y=-5\\hfill \\\\ \\,\\,\\,3x-2y=-12\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1698278\">[latex]\\left(-2,3\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1671248\">\n<div id=\"fs-id1671249\">\n<p id=\"fs-id1671250\">[latex]\\begin{array}{l}\\frac{4}{7}x+\\frac{1}{5}y=\\frac{43}{70}\\\\ \\frac{5}{6}x-\\frac{1}{3}y=-\\frac{2}{3}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1523916\">\n<div id=\"fs-id1523917\">\n<p id=\"fs-id1523918\">[latex]\\begin{array}{l}5x+6y=14\\\\ 4x+8y=8\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1428954\">[latex]\\left(4,-1\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1425491\">For the following exercises, use addition to solve the system of equations.<\/p>\n<div id=\"fs-id1425494\">\n<div id=\"fs-id1425495\">\n<p id=\"fs-id1425496\">[latex]\\begin{array}{l}3x+2y=-7\\\\ 2x+4y=6\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1512233\">\n<div id=\"fs-id1512234\">\n<p id=\"fs-id1512235\">[latex]\\begin{array}{r}3x+4y=2\\\\ 9x+12y=3\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1584732\">No solutions exist.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1584736\">\n<div id=\"fs-id1584737\">\n<p id=\"fs-id1584738\">[latex]\\begin{array}{l}8x+4y=2\\\\ 6x-5y=0.7\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1459641\">For the following exercises, write a system of equations to solve each problem. Solve the system of equations.<\/p>\n<div id=\"fs-id1459646\">\n<div id=\"fs-id1459647\">\n<p id=\"fs-id1459648\">A factory has a cost of production[latex]\\,C\\left(x\\right)=150x+15\\text{,}000\\,[\/latex]and a revenue function[latex]\\,R\\left(x\\right)=200x.\\,[\/latex]What is the break-even point?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1536207\">[latex]\\left(300,60,000\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1423656\">\n<div id=\"fs-id1423657\">\n<p id=\"fs-id1423658\">A performer charges[latex]\\,C\\left(x\\right)=50x+10\\text{,}000,\\,[\/latex]where[latex]\\,x\\,[\/latex]is the total number of attendees at a show. The venue charges $75 per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1358367\">[latex]\\left(400,30,000\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2266991\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/4f2b4755-b59c-4f4f-b4c3-e97938c93fcd\">Systems of Linear Equations: Three Variables<\/a><\/h4>\n<p id=\"fs-id1300053\">For the following exercises, solve the system of three equations using substitution or addition.<\/p>\n<div id=\"fs-id1512118\">\n<div id=\"fs-id1512119\">\n<p id=\"fs-id1512120\">[latex]\\begin{array}{l}\\text{ }0.5x-0.5y=10\\hfill \\\\ \\text{ }-0.2y+0.2x=4\\hfill \\\\ \\text{ }0.1x+0.1z=2\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1513414\">[latex]\\left(10,-10,10\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1705168\">\n<div id=\"fs-id1705169\">\n<p id=\"fs-id1705170\">[latex]\\begin{array}{r}\\hfill 5x+3y-z=5\\,\\,\\,\\\\ \\hfill 3x-2y+4z=13\\\\ \\hfill 4x+3y+5z=22\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1524026\">\n<div id=\"fs-id1524027\">\n<p id=\"fs-id1524028\">[latex]\\begin{array}{r}x+y+z=1\\\\ 2x+2y+2z=1\\\\ 3x+3y=2\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1366211\">No solutions exist.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2633372\">\n<div id=\"fs-id2633373\">\n<p id=\"fs-id2633374\">[latex]\\begin{array}{l}\\text{ }2x-3y+z=-1\\hfill \\\\ \\text{ }x+y+z=-4\\hfill \\\\ \\text{ }4x+2y-3z=33\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1677098\">\n<div id=\"fs-id1677099\">\n<p id=\"fs-id1677100\">[latex]\\begin{array}{l}\\,\\,3x+2y-z=-10\\hfill \\\\ \\,\\,\\,\\,x-y+2z=7\\hfill \\\\ -x+3y+z=-2\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1409562\">[latex]\\left(-1,-2,3\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1406449\">\n<div id=\"fs-id1406450\">\n<p id=\"fs-id1406451\">[latex]\\begin{array}{r}\\hfill 3x+4z=-11\\\\ \\hfill x-2y=5\\,\\,\\,\\,\\,\\,\\,\\\\ \\hfill 4y-z=-10\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1417289\">\n<div id=\"fs-id1417290\">\n<p id=\"fs-id1417291\">[latex]\\begin{array}{r}2x-3y+z=0\\\\ 2x+4y-3z=0\\\\ 6x-2y-z=0\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1417064\">[latex]\\left(x,\\frac{8x}{5},\\frac{14x}{5}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1532027\">\n<div id=\"fs-id1532028\">\n<p id=\"fs-id1532029\">[latex]\\begin{array}{r}6x-4y-2z=2\\\\ 3x+2y-5z=4\\\\ 6y-7z=5\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1587443\">For the following exercises, write a system of equations to solve each problem. Solve the system of equations.<\/p>\n<div id=\"fs-id1587447\">\n<div id=\"fs-id1706452\">\n<p id=\"fs-id1706453\">Three odd numbers sum up to 61. The smaller is one-third the larger and the middle number is 16 less than the larger. What are the three numbers?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1706459\">11, 17, 33<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1706464\">\n<div id=\"fs-id1706465\">\n<p id=\"fs-id1706466\">A local theatre sells out for their show. They sell all 500 tickets for a total purse of $8,070.00. The tickets were priced at $15 for students, $12 for children, and $18 for adults. If the band sold three times as many adult tickets as children\u2019s tickets, how many of each type was sold?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1801019\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/228bf1c6-e200-41b3-9bc0-79b6f6edbb37\">Systems of Nonlinear Equations and Inequalities: Two Variables<\/a><\/h4>\n<p id=\"fs-id1706479\">For the following exercises, solve the system of nonlinear equations.<\/p>\n<div id=\"fs-id1584246\">\n<div id=\"fs-id1584247\">\n<p id=\"fs-id1584248\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ y={x}^{2}-7\\end{array}\\hfill \\\\ y=5x-13\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1695047\">[latex]\\left(2,-3\\right),\\left(3,2\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1403228\">\n<div id=\"fs-id1403229\">\n<p id=\"fs-id1403230\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ y={x}^{2}-4\\end{array}\\hfill \\\\ y=5x+10\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1434420\">\n<div id=\"fs-id1434421\">\n<p id=\"fs-id1434422\">[latex]\\begin{array}{l}{x}^{2}+{y}^{2}=16\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y=x-8\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1403141\">No solution<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1403146\">\n<div id=\"fs-id1403147\">\n<p id=\"fs-id1403148\">[latex]\\begin{array}{l}{x}^{2}+{y}^{2}=25\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y={x}^{2}+5\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1451621\">\n<div id=\"fs-id1451622\">\n<p id=\"fs-id1451623\">[latex]\\begin{array}{r}{x}^{2}+{y}^{2}=4\\\\ y-{x}^{2}=3\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1513095\">No solution<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1513100\">For the following exercises, graph the inequality.<\/p>\n<div id=\"fs-id1513103\">\n<div id=\"fs-id1513104\">\n<p id=\"fs-id1513105\">[latex]y>{x}^{2}-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1457141\">\n<div id=\"fs-id1457142\">\n<p id=\"fs-id1457143\">[latex]\\frac{1}{4}{x}^{2}+{y}^{2}<4[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1663829\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154215\/CNX_Precalc_Figure_09_08_202.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1663839\">For the following exercises, graph the system of inequalities.<\/p>\n<div id=\"fs-id1663842\">\n<div id=\"fs-id1663844\">\n<p id=\"fs-id1663845\">[latex]\\begin{array}{l}{x}^{2}+{y}^{2}+2x<3\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y>-{x}^{2}-3\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1646717\">\n<div id=\"fs-id1646718\">\n<p id=\"fs-id1646719\">[latex]\\begin{array}{l}{x}^{2}-2x+{y}^{2}-4x<4\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y<-x+4\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1600817\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154218\/CNX_Precalc_Figure_09_08_204.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1600828\">\n<div id=\"fs-id1600829\">\n<p id=\"fs-id1600830\">[latex]\\begin{array}{l}{x}^{2}+{y}^{2}<1\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{y}^{2}<x\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2867422\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/b61e8ccf-4f28-453e-9e41-99b6b7c8dfe8\">Partial Fractions<\/a><\/h4>\n<p id=\"fs-id1409014\">For the following exercises, decompose into partial fractions.<\/p>\n<div>\n<div id=\"fs-id1409018\">\n<p id=\"fs-id1409019\">[latex]\\frac{-2x+6}{{x}^{2}+3x+2}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1530172\">[latex]\\frac{2}{x+2},\\frac{-4}{x+1}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1430683\">\n<div id=\"fs-id1430684\">\n<p id=\"fs-id1430685\">[latex]\\frac{10x+2}{4{x}^{2}+4x+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1564134\">\n<div id=\"fs-id1564135\">\n<p id=\"fs-id1564136\">[latex]\\frac{7x+20}{{x}^{2}+10x+25}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1440535\">[latex]\\frac{7}{x+5},\\frac{-15}{{\\left(x+5\\right)}^{2}}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1614973\">\n<div id=\"fs-id1614974\">\n<p id=\"fs-id1614975\">[latex]\\frac{x-18}{{x}^{2}-12x+36}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1417354\">\n<div id=\"fs-id1417355\">\n<p id=\"fs-id1417356\">[latex]\\frac{-{x}^{2}+36x+70}{{x}^{3}-125}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1528465\">[latex]\\frac{3}{x-5},\\frac{-4x+1}{{x}^{2}+5x+25}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1583086\">\n<div id=\"fs-id1583087\">\n<p id=\"fs-id1583088\">[latex]\\frac{-5{x}^{2}+6x-2}{{x}^{3}+27}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1431813\">\n<div id=\"fs-id1431814\">\n<p id=\"fs-id1431815\">[latex]\\frac{{x}^{3}-4{x}^{2}+3x+11}{{\\left({x}^{2}-2\\right)}^{2}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1582588\">[latex]\\frac{x-4}{\\left({x}^{2}-2\\right)},\\frac{5x+3}{{\\left({x}^{2}-2\\right)}^{2}}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1655916\">\n<div id=\"fs-id1655917\">\n<p id=\"fs-id1655918\">[latex]\\frac{4{x}^{4}-2{x}^{3}+22{x}^{2}-6x+48}{x{\\left({x}^{2}+4\\right)}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2925191\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/508a4d4e-0136-4c6c-87b8-e210022d69b4\">Matrices and Matrix Operations<\/a><\/h4>\n<p id=\"fs-id1581865\">For the following exercises, perform the requested operations on the given matrices.<\/p>\n<div id=\"fs-id1581869\">[latex]A=\\left[\\begin{array}{rr}\\hfill 4& \\hfill -2\\\\ \\hfill 1& \\hfill 3\\end{array}\\right],B=\\left[\\begin{array}{rrr}\\hfill 6& \\hfill 7& \\hfill -3\\\\ \\hfill 11& \\hfill -2& \\hfill 4\\end{array}\\right],C=\\left[\\begin{array}{r}\\hfill \\begin{array}{cc}6& 7\\\\ 11& -2\\end{array}\\\\ \\hfill \\begin{array}{cc}14& 0\\end{array}\\end{array}\\right],D=\\left[\\begin{array}{rrr}\\hfill 1& \\hfill -4& \\hfill 9\\\\ \\hfill 10& \\hfill 5& \\hfill -7\\\\ \\hfill 2& \\hfill 8& \\hfill 5\\end{array}\\right],E=\\left[\\begin{array}{rrr}\\hfill 7& \\hfill -14& \\hfill 3\\\\ \\hfill 2& \\hfill -1& \\hfill 3\\\\ \\hfill 0& \\hfill 1& \\hfill 9\\end{array}\\right][\/latex]<\/div>\n<div id=\"fs-id1519664\">\n<div id=\"fs-id1519665\">\n<p id=\"fs-id1519666\">[latex]-4A[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1615553\">[latex]\\left[\\begin{array}{cc}-16& 8\\\\ -4& -12\\end{array}\\right][\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1535716\">\n<div id=\"fs-id1535717\">\n<p id=\"fs-id1535718\">[latex]10D-6E[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1536793\">\n<div id=\"fs-id1536794\">\n<p id=\"fs-id1536796\">[latex]B+C[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1706172\">undefined; dimensions do not match<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1706176\">\n<div id=\"fs-id1706177\">\n<p id=\"fs-id1706178\">[latex]AB[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1650861\">\n<div id=\"fs-id1650862\">\n<p id=\"fs-id1650863\">[latex]BA[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1650879\">undefined; inner dimensions do not match<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1650884\">\n<div id=\"fs-id1650885\">\n<p id=\"fs-id1650886\">[latex]BC[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1647717\">\n<div id=\"fs-id1647718\">\n<p id=\"fs-id1647720\">[latex]CB[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1647736\">[latex]\\left[\\begin{array}{ccc}113& 28& 10\\\\ 44& 81& -41\\\\ 84& 98& -42\\end{array}\\right][\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1419710\">\n<div id=\"fs-id1419711\">\n<p id=\"fs-id1419712\">[latex]DE[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1419726\">\n<div id=\"fs-id1419727\">\n<p id=\"fs-id1419728\">[latex]ED[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1693755\">[latex]\\left[\\begin{array}{ccc}-127& -74& 176\\\\ -2& 11& 40\\\\ 28& 77& 38\\end{array}\\right][\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1366518\">\n<div id=\"fs-id1366519\">\n<p id=\"fs-id1366520\">[latex]EC[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1366534\">\n<div id=\"fs-id1366535\">\n<p id=\"fs-id1366536\">[latex]CE[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1700302\">undefined; inner dimensions do not match<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1700306\">\n<div id=\"fs-id1700307\">\n<p id=\"fs-id1700308\">[latex]{A}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2652002\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/69e0b9a7-0928-46a6-bb59-6cd28f18eec9\">Solving Systems with Gaussian Elimination<\/a><\/h4>\n<p id=\"fs-id1407521\">For the following exercises, write the system of linear equations from the augmented matrix. Indicate whether there will be a unique solution.<\/p>\n<div id=\"fs-id1407525\">\n<div id=\"fs-id1407526\">\n<p id=\"fs-id1407528\">[latex]\\left[\\begin{array}{rrr}\\hfill 1& \\hfill 0& \\hfill -3\\\\ \\hfill 0& \\hfill 1& \\hfill 2\\\\ \\hfill 0& \\hfill 0& \\hfill 0\\end{array}\\text{ }|\\text{ }\\begin{array}{r}\\hfill 7\\\\ \\hfill -5\\\\ \\hfill 0\\end{array}\\right][\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1637532\">[latex]\\begin{array}{l}x-3z=7\\\\ y+2z=-5\\,\\end{array}[\/latex]with infinite solutions<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1523479\">\n<div>\n<p id=\"fs-id1523481\">[latex]\\left[\\begin{array}{rrr}\\hfill 1& \\hfill 0& \\hfill 5\\\\ \\hfill 0& \\hfill 1& \\hfill -2\\\\ \\hfill 0& \\hfill 0& \\hfill 0\\end{array}\\text{ }|\\text{ }\\begin{array}{r}\\hfill -9\\\\ \\hfill 4\\\\ \\hfill 3\\end{array}\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1375129\">For the following exercises, write the augmented matrix from the system of linear equations.<\/p>\n<div id=\"fs-id1375132\">\n<div id=\"fs-id1375133\">\n<p id=\"fs-id1375134\">[latex]\\begin{array}{l}\\\\ \\begin{array}{r}\\hfill -2x+2y+z=7\\\\ \\hfill 2x-8y+5z=0\\\\ \\hfill 19x-10y+22z=3\\end{array}\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1656057\">[latex]\\left[\\begin{array}{rrr}\\hfill -2& \\hfill 2& \\hfill 1\\\\ \\hfill 2& \\hfill -8& \\hfill 5\\\\ \\hfill 19& \\hfill -10& \\hfill 22\\end{array}\\text{ }|\\text{ }\\begin{array}{r}\\hfill 7\\\\ \\hfill 0\\\\ \\hfill 3\\end{array}\\right][\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1452352\">\n<div id=\"fs-id1452353\">\n<p id=\"fs-id1452354\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,4x+2y-3z=14\\hfill \\\\ -12x+3y+z=100\\hfill \\\\ \\,\\,\\,\\,\\,9x-6y+2z=31\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1457569\">\n<div id=\"fs-id1457570\">\n<p id=\"fs-id1457571\">[latex]\\begin{array}{r}\\hfill x+3z=12\\,\\\\ \\hfill -x+4y=0\\,\\,\\,\\,\\\\ \\hfill y+2z=-7\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1528512\">[latex]\\left[\\begin{array}{rrr}\\hfill 1& \\hfill 0& \\hfill 3\\\\ \\hfill -1& \\hfill 4& \\hfill 0\\\\ \\hfill 0& \\hfill 1& \\hfill 2\\end{array}\\text{ }|\\text{ }\\begin{array}{r}\\hfill 12\\\\ \\hfill 0\\\\ \\hfill -7\\end{array}\\right][\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1601028\">For the following exercises, solve the system of linear equations using Gaussian elimination.<\/p>\n<div id=\"fs-id1410109\">\n<div id=\"fs-id1410110\">\n<p id=\"fs-id1410111\">[latex]\\begin{array}{r}3x-4y=-7\\\\ -6x+8y=14\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1626545\">\n<div id=\"fs-id1626546\">\n<p id=\"fs-id1626548\">[latex]\\begin{array}{r}3x-4y=1\\\\ -6x+8y=6\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1422121\">No solutions exist.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1503778\">\n<div id=\"fs-id1503779\">\n<p id=\"fs-id1503780\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ -1.1x-2.3y=6.2\\end{array}\\hfill \\\\ -5.2x-4.1y=4.3\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1436195\">\n<div id=\"fs-id1436196\">\n<p id=\"fs-id1436197\">[latex]\\begin{array}{r}\\hfill 2x+3y+2z=1\\,\\,\\,\\,\\,\\\\ \\hfill -4x-6y-4z=-2\\\\ \\hfill 10x+15y+10z=0\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1455902\">No solutions exist.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1455907\">\n<div id=\"fs-id1455908\">\n<p id=\"fs-id1455909\">[latex]\\begin{array}{r}\\hfill -x+2y-4z=8\\,\\,\\,\\,\\\\ \\hfill 3y+8z=-4\\\\ \\hfill -7x+y+2z=1\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1766016\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/634d2387-7429-480f-a04e-867c2f9699fb\">Solving Systems with Inverses<\/a><\/h4>\n<p id=\"fs-id1431604\">For the following exercises, find the inverse of the matrix.<\/p>\n<div id=\"fs-id1431608\">\n<div id=\"fs-id1431609\">\n<p id=\"fs-id1431610\">[latex]\\left[\\begin{array}{rr}\\hfill -0.2& \\hfill 1.4\\\\ \\hfill 1.2& \\hfill -0.4\\end{array}\\right][\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1531734\">[latex]\\frac{1}{8}\\left[\\begin{array}{cc}2& 7\\\\ 6& 1\\end{array}\\right][\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1949652\">\n<div id=\"fs-id1949653\">\n<p id=\"fs-id1949654\">[latex]\\left[\\begin{array}{rr}\\hfill \\frac{1}{2}& \\hfill -\\frac{1}{2}\\\\ \\hfill -\\frac{1}{4}& \\hfill \\frac{3}{4}\\end{array}\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1456514\">\n<div id=\"fs-id1456515\">\n<p id=\"fs-id1456516\">[latex]\\left[\\begin{array}{ccc}12& 9& -6\\\\ -1& 3& 2\\\\ -4& -3& 2\\end{array}\\right][\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1514199\">No inverse exists.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1514203\">\n<div id=\"fs-id1514204\">\n<p id=\"fs-id1514205\">[latex]\\left[\\begin{array}{ccc}2& 1& 3\\\\ 1& 2& 3\\\\ 3& 2& 1\\end{array}\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1507401\">For the following exercises, find the solutions by computing the inverse of the matrix.<\/p>\n<div id=\"fs-id1507405\">\n<div id=\"fs-id1507406\">\n<p id=\"fs-id1507407\">[latex]\\begin{array}{l}\\,\\,\\,\\,0.3x-0.1y=-10\\hfill \\\\ -0.1x+0.3y=14\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id299169\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id299171\">[latex]\\left(-20,40\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1601516\">\n<div id=\"fs-id1601518\">\n<p id=\"fs-id1601519\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,0.4x-0.2y=-0.6\\hfill \\\\ -0.1x+0.05y=0.3\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1584782\">\n<div id=\"fs-id1584783\">\n<p id=\"fs-id1584784\">[latex]\\begin{array}{r}4x+3y-3z=-4.3\\\\ 5x-4y-z=-6.1\\\\ x+z=-0.7\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1531936\">[latex]\\left(-1,0.2,0.3\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1519539\">\n<div id=\"fs-id1519540\">\n<p id=\"fs-id1519542\">[latex]\\begin{array}{r}\\hfill \\begin{array}{l}\\\\ -2x-3y+2z=3\\end{array}\\\\ \\hfill -x+2y+4z=-5\\\\ \\hfill -2y+5z=-3\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1403397\">For the following exercises, write a system of equations to solve each problem. Solve the system of equations.<\/p>\n<div id=\"fs-id1403401\">\n<div id=\"fs-id1403402\">\n<p id=\"fs-id1403403\">Students were asked to bring their favorite fruit to class. 90% of the fruits consisted of banana, apple, and oranges. If oranges were half as popular as bananas and apples were 5% more popular than bananas, what are the percentages of each individual fruit?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1403411\">17% oranges, 34% bananas, 39% apples<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1403415\">\n<div id=\"fs-id1403416\">\n<p id=\"fs-id1403417\">A sorority held a bake sale to raise money and sold brownies and chocolate chip cookies. They priced the brownies at $2 and the chocolate chip cookies at $1. They raised $250 and sold 175 items. How many brownies and how many cookies were sold?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2974900\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/e75dadb0-5d43-42f2-8e78-0ad237f10096\">Solving Systems with Cramer&#8217;s Rule<\/a><\/h4>\n<p id=\"fs-id1357039\">For the following exercises, find the determinant.<\/p>\n<div id=\"fs-id1357042\">\n<div id=\"fs-id1357043\">\n<p id=\"fs-id1357044\">[latex]|\\begin{array}{cc}100& 0\\\\ 0& 0\\end{array}|[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1504600\">0<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1504604\">\n<div id=\"fs-id1504605\">\n<p id=\"fs-id1504606\">[latex]|\\begin{array}{cc}0.2& -0.6\\\\ 0.7& -1.1\\end{array}|[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1535261\">\n<div id=\"fs-id1535262\">\n<p id=\"fs-id1535263\">[latex]|\\begin{array}{ccc}-1& 4& 3\\\\ 0& 2& 3\\\\ 0& 0& -3\\end{array}|[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1698281\">6<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1698286\">\n<div id=\"fs-id1698287\">\n<p id=\"fs-id1698288\">[latex]|\\begin{array}{ccc}\\sqrt{2}& 0& 0\\\\ 0& \\sqrt{2}& 0\\\\ 0& 0& \\sqrt{2}\\end{array}|[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1534786\">For the following exercises, use Cramer\u2019s Rule to solve the linear systems of equations.<\/p>\n<div id=\"fs-id1534791\">\n<div id=\"fs-id1534792\">\n<p id=\"fs-id1534793\">[latex]\\begin{array}{r}\\hfill 4x-2y=23\\,\\,\\,\\,\\\\ \\hfill -5x-10y=-35\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1508832\">[latex]\\left(6,\\frac{1}{2}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1698540\">\n<div id=\"fs-id1698541\">\n<p id=\"fs-id1698542\">[latex]\\begin{array}{l}0.2x-0.1y=0\\\\ -0.3x+0.3y=2.5\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1689845\">\n<div id=\"fs-id1689846\">\n<p id=\"fs-id1689847\">[latex]\\begin{array}{r}\\hfill -0.5x+0.1y=0.3\\,\\,\\,\\\\ \\hfill -0.25x+0.05y=0.15\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1584180\">(<em>x<\/em>, 5<em>x <\/em>+ 3)<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1584193\">\n<div id=\"fs-id1508743\">\n<p id=\"fs-id1508744\">[latex]\\begin{array}{l}x+6y+3z=4\\\\ 2x+y+2z=3\\\\ 3x-2y+z=0\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1676826\">\n<div id=\"fs-id1676827\">\n<p id=\"fs-id1676828\">[latex]\\begin{array}{r}\\hfill 4x-3y+5z=-\\frac{5}{2}\\\\ \\hfill 7x-9y-3z=\\frac{3}{2}\\,\\,\\,\\,\\\\ \\hfill x-5y-5z=\\frac{5}{2}\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1508865\">[latex]\\left(0,0,-\\frac{1}{2}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1658138\">\n<div id=\"fs-id1658139\">\n<p id=\"fs-id1658140\">[latex]\\begin{array}{r}\\frac{3}{10}x-\\frac{1}{5}y-\\frac{3}{10}z=-\\frac{1}{50}\\\\ \\frac{1}{10}x-\\frac{1}{10}y-\\frac{1}{2}z=-\\frac{9}{50}\\\\ \\frac{2}{5}x-\\frac{1}{2}y-\\frac{3}{5}z=-\\frac{1}{5}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1459744\" class=\"practice-test\">\n<h3>Practice Test<\/h3>\n<p id=\"fs-id1459748\">Is the following ordered pair a solution to the system of equations?<\/p>\n<div id=\"fs-id1459751\">\n<div id=\"fs-id1459752\">\n<p id=\"fs-id1459753\">[latex]\\begin{array}{l}\\\\ \\begin{array}{l}-5x-y=12\\,\\hfill \\\\ x+4y=9\\hfill \\end{array}\\end{array}[\/latex]with[latex]\\,\\left(-3,3\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1435942\">Yes<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1403240\">For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.<\/p>\n<div id=\"fs-id1403244\">\n<div id=\"fs-id1403246\">\n<p id=\"fs-id1403247\">[latex]\\begin{array}{r}\\frac{1}{2}x-\\frac{1}{3}y=4\\\\ \\frac{3}{2}x-y=0\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1406347\">\n<div id=\"fs-id1406348\">\n<p id=\"fs-id1406349\">[latex]\\begin{array}{r}\\hfill \\begin{array}{l}\\\\ -\\frac{1}{2}x-4y=4\\end{array}\\\\ \\hfill 2x+16y=2\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1406246\">No solutions exist.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1406251\">\n<div id=\"fs-id1406252\">\n<p id=\"fs-id1406253\">[latex]\\begin{array}{r}\\hfill 5x-y=1\\,\\,\\,\\,\\\\ \\hfill -10x+2y=-2\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1780639\">\n<div id=\"fs-id1780640\">\n<p id=\"fs-id1780641\">[latex]\\begin{array}{l}4x-6y-2z=\\frac{1}{10}\\hfill \\\\ \\,\\,\\,x-7y+5z=-\\frac{1}{4}\\hfill \\\\ 3x+6y-9z=\\frac{6}{5}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1455840\">[latex]\\frac{1}{20}\\left(10,5,4\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1675200\">\n<div id=\"fs-id1675201\">\n<p id=\"fs-id1675202\">[latex]\\begin{array}{r}x+z=20\\\\ x+y+z=20\\\\ x+2y+z=10\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1436662\">\n<div id=\"fs-id1436663\">\n<p id=\"fs-id1436664\">[latex]\\begin{array}{r}5x-4y-3z=0\\\\ 2x+y+2z=0\\\\ x-6y-7z=0\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1615810\">[latex]\\left(x,\\frac{16x}{5}-\\frac{13x}{5}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1537027\">\n<div id=\"fs-id1537028\">\n<p id=\"fs-id1537029\">[latex]\\begin{array}{l}y={x}^{2}+2x-3\\\\ y=x-1\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1432229\">\n<div id=\"fs-id1432230\">\n<p id=\"fs-id1432231\">[latex]\\begin{array}{l}{y}^{2}+{x}^{2}=25\\\\ {y}^{2}-2{x}^{2}=1\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1703248\">[latex]\\left(-2\\sqrt{2},-\\sqrt{17}\\right),\\left(-2\\sqrt{2},\\sqrt{17}\\right),\\left(2\\sqrt{2},-\\sqrt{17}\\right),\\left(2\\sqrt{2},\\sqrt{17}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1384560\">For the following exercises, graph the following inequalities.<\/p>\n<div id=\"fs-id1384563\">\n<div id=\"fs-id1384564\">\n<p id=\"fs-id1384565\">[latex]y<{x}^{2}+9[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1536358\">\n<div id=\"fs-id1536359\">\n<p id=\"fs-id1536360\">[latex]\\begin{array}{l}{x}^{2}+{y}^{2}>4\\\\ y<{x}^{2}+1\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1380141\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154228\/CNX_Precalc_Figure_09_08_207.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1663752\">For the following exercises, write the partial fraction decomposition.<\/p>\n<div id=\"fs-id1663755\">\n<div id=\"fs-id1663756\">\n<p id=\"fs-id1663757\">[latex]\\frac{-8x-30}{{x}^{2}+10x+25}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1507242\">\n<div id=\"fs-id1507243\">\n<p id=\"fs-id1598618\">[latex]\\frac{13x+2}{{\\left(3x+1\\right)}^{2}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1519238\">[latex]\\frac{5}{3x+1}-\\frac{2x+3}{{\\left(3x+1\\right)}^{2}}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1698396\">\n<div id=\"fs-id1698397\">\n<p id=\"fs-id1698398\">[latex]\\frac{{x}^{4}-{x}^{3}+2x-1}{x{\\left({x}^{2}+1\\right)}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1675019\">For the following exercises, perform the given matrix operations.<\/p>\n<div id=\"fs-id1675022\">\n<div id=\"fs-id1675024\">\n<p id=\"fs-id1675025\">[latex]5\\left[\\begin{array}{cc}4& 9\\\\ -2& 3\\end{array}\\right]+\\frac{1}{2}\\left[\\begin{array}{cc}-6& 12\\\\ 4& -8\\end{array}\\right][\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1694107\">[latex]\\left[\\begin{array}{cc}17& 51\\\\ -8& 11\\end{array}\\right][\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1527496\">\n<div id=\"fs-id1527497\">\n<p id=\"fs-id1527498\">[latex]\\left[\\begin{array}{rrr}\\hfill 1& \\hfill 4& \\hfill -7\\\\ \\hfill -2& \\hfill 9& \\hfill 5\\\\ \\hfill 12& \\hfill 0& \\hfill -4\\end{array}\\right]\\text{ }\\left[\\begin{array}{cc}3& -4\\\\ 1& 3\\\\ 5& 10\\end{array}\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1392877\">\n<div id=\"fs-id1392878\">\n<p id=\"fs-id1392879\">[latex]{\\left[\\begin{array}{rr}\\hfill \\frac{1}{2}& \\hfill \\frac{1}{3}\\\\ \\hfill \\frac{1}{4}& \\hfill \\frac{1}{5}\\end{array}\\right]}^{-1}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1583988\">[latex]\\left[\\begin{array}{cc}12& -20\\\\ -15& 30\\end{array}\\right][\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1644580\">\n<div id=\"fs-id1644581\">\n<p id=\"fs-id1644582\">[latex]\\mathrm{det}|\\begin{array}{cc}0& 0\\\\ 400& 4\\text{,}000\\end{array}|[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1648258\">\n<div id=\"fs-id1648259\">\n<p id=\"fs-id1648260\">[latex]\\mathrm{det}|\\begin{array}{rrr}\\hfill \\frac{1}{2}& \\hfill -\\frac{1}{2}& \\hfill 0\\\\ \\hfill -\\frac{1}{2}& \\hfill 0& \\hfill \\frac{1}{2}\\\\ \\hfill 0& \\hfill \\frac{1}{2}& \\hfill 0\\end{array}|[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1295612\">[latex]-\\frac{1}{8}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1615192\">\n<div id=\"fs-id1615193\">\n<p id=\"fs-id1615194\">If[latex]\\,\\mathrm{det}\\left(A\\right)=-6,\\,[\/latex] what would be the determinant if you switched rows 1 and 3, multiplied the second row by 12, and took the inverse?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1595084\">\n<div id=\"fs-id1595085\">\n<p id=\"fs-id1595086\">Rewrite the system of linear equations as an augmented matrix.<\/p>\n<div id=\"fs-id1595090\">[latex]\\begin{array}{l}14x-2y+13z=140\\hfill \\\\ -2x+3y-6z=-1\\hfill \\\\ x-5y+12z=11\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1456561\">[latex]\\left[\\begin{array}{rrr}\\hfill 14& \\hfill -2& \\hfill 13\\\\ \\hfill -2& \\hfill 3& \\hfill -6\\\\ \\hfill 1& \\hfill -5& \\hfill 12\\end{array}\\text{ }|\\text{ }\\begin{array}{r}\\hfill 140\\\\ \\hfill -1\\\\ \\hfill 11\\end{array}\\right][\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1599476\">\n<div id=\"fs-id1599477\">\n<p id=\"fs-id1599478\">Rewrite the augmented matrix as a system of linear equations.<\/p>\n<div id=\"fs-id1599481\">[latex]\\left[\\begin{array}{rrr}\\hfill 1& \\hfill 0& \\hfill 3\\\\ \\hfill -2& \\hfill 4& \\hfill 9\\\\ \\hfill -6& \\hfill 1& \\hfill 2\\end{array}|\\begin{array}{r}\\hfill 12\\\\ \\hfill -5\\\\ \\hfill 8\\end{array}\\right][\/latex]<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1636943\">For the following exercises, use Gaussian elimination to solve the systems of equations.<\/p>\n<div id=\"fs-id1636946\">\n<div id=\"fs-id1636947\">\n<p id=\"fs-id1636948\">[latex]\\begin{array}{r}x-6y=4\\\\ 2x-12y=0\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1518911\">No solutions exist.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1518915\">\n<div id=\"fs-id1518916\">\n<p id=\"fs-id1518917\">[latex]\\begin{array}{r}\\hfill 2x+y+z=-3\\\\ \\hfill x-2y+3z=6\\,\\,\\,\\,\\\\ \\hfill x-y-z=6\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1699601\">For the following exercises, use the inverse of a matrix to solve the systems of equations.<\/p>\n<div id=\"fs-id1699604\">\n<div id=\"fs-id1699605\">\n<p id=\"fs-id1699606\">[latex]\\begin{array}{r}\\hfill 4x-5y=-50\\\\ \\hfill -x+2y=80\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1440484\">[latex]\\left(100,90\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1568663\">\n<div id=\"fs-id1568664\">\n<p id=\"fs-id1568666\">[latex]\\begin{array}{r}\\hfill \\frac{1}{100}x-\\frac{3}{100}y+\\frac{1}{20}z=-49\\\\ \\hfill \\frac{3}{100}x-\\frac{7}{100}y-\\frac{1}{100}z=13\\,\\,\\,\\,\\\\ \\hfill \\frac{9}{100}x-\\frac{9}{100}y-\\frac{9}{100}z=99\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1653924\">For the following exercises, use Cramer\u2019s Rule to solve the systems of equations.<\/p>\n<div id=\"fs-id1653928\">\n<div id=\"fs-id1653929\">\n<p id=\"fs-id1653930\">[latex]\\begin{array}{l}200x-300y=2\\\\ 400x+715y=4\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1703905\">[latex]\\left(\\frac{1}{100},0\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1527386\">\n<div id=\"fs-id1527387\">\n<p id=\"fs-id1527388\">[latex]\\begin{array}{l}0.1x+0.1y-0.1z=-1.2\\\\ 0.1x-0.2y+0.4z=-1.2\\\\ 0.5x-0.3y+0.8z=-5.9\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1528262\">For the following exercises, solve using a system of linear equations.<\/p>\n<div id=\"fs-id1528266\">\n<div id=\"fs-id1528267\">\n<p id=\"fs-id1528268\">A factory producing cell phones has the following cost and revenue functions:[latex]\\,C\\left(x\\right)={x}^{2}+75x+2\\text{,}688\\,[\/latex]and[latex]\\,R\\left(x\\right)={x}^{2}+160x.\\,[\/latex]What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1598123\">32 or more cell phones per day<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1598127\">\n<div id=\"fs-id1598128\">\n<p id=\"fs-id1598129\">A small fair charges $1.50 for students, $1 for children, and $2 for adults. In one day, three times as many children as adults attended. A total of 800 tickets were sold for a total revenue of $1,050. How many of each type of ticket was sold?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1674058\">\n<dt>Cramer\u2019s Rule<\/dt>\n<dd id=\"fs-id1674063\">a method for solving systems of equations that have the same number of equations as variables using determinants<\/dd>\n<\/dl>\n<dl id=\"fs-id1674068\">\n<dt>determinant<\/dt>\n<dd id=\"fs-id1674074\">a number calculated using the entries of a square matrix that determines such information as whether there is a solution to a system of equations<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":291,"menu_order":9,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-183","chapter","type-chapter","status-publish","hentry"],"part":166,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/183","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/users\/291"}],"version-history":[{"count":1,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/183\/revisions"}],"predecessor-version":[{"id":184,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/183\/revisions\/184"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/parts\/166"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/183\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=183"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=183"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=183"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=183"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}