{"id":2513,"date":"2018-12-11T13:41:37","date_gmt":"2018-12-11T18:41:37","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-systems-of-equations-with-three-variables\/"},"modified":"2018-12-11T13:41:37","modified_gmt":"2018-12-11T18:41:37","slug":"solve-systems-of-equations-with-three-variables","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-systems-of-equations-with-three-variables\/","title":{"raw":"Solve Systems of Equations with Three Variables","rendered":"Solve Systems of Equations with Three Variables"},"content":{"raw":"\n[latexpage]<div class=\"textbox textbox--learning-objectives\"><h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>By the end of this section, you will be able to: <ul><li>Determine whether an ordered triple is a solution of a system of three linear equations with three variables<\/li><li>Solve a system of linear equations with three variables<\/li><li>Solve applications using systems of linear equations with three variables<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167829709277\" class=\"be-prepared\"><p id=\"fs-id1167833311246\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167833055966\" type=\"1\"><li>Evaluate \\(5x-2y+3z\\) when \\(x=-2,\\) \\(y=-4,\\) and \\(z=3.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167832053133\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Classify the equations as a conditional equation, an identity, or a contradiction and then state the solution. \\(\\left\\{\\begin{array}{c}-2x+y=-11\\hfill \\\\ x+3y=9\\hfill \\end{array}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167836666645\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Classify the equations as a conditional equation, an identity, or a contradiction and then state the solution. \\(\\left\\{\\begin{array}{c}7x+8y=4\\hfill \\\\ 3x-5y=27\\hfill \\end{array}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167826211749\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836729743\"><h3 data-type=\"title\">Determine Whether an Ordered Triple is a Solution of a System of Three Linear Equations with Three Variables<\/h3><p id=\"fs-id1167836543656\">In this section, we will extend our work of solving a system of linear equations. So far we have worked with <span data-type=\"term\" class=\"no-emphasis\">systems of equations<\/span> with two equations and two variables. Now we will work with systems of three equations with three variables. But first let's review what we already know about solving equations and systems involving up to two variables.<\/p><p id=\"fs-id1167836542288\">We learned earlier that the graph of a <span data-type=\"term\" class=\"no-emphasis\">linear equation<\/span>, \\(ax+by=c,\\) is a line. Each point on the line, an ordered pair \\(\\left(x,y\\right),\\) is a solution to the equation. For a system of two equations with two variables, we graph two lines. Then we can see that all the points that are solutions to each equation form a line. And, by finding what the lines have in common, we\u2019ll find the solution to the system.<\/p><p id=\"fs-id1167829690735\">Most linear equations in one variable have one solution, but we saw that some equations, called contradictions, have no solutions and for other equations, called identities, all numbers are solutions<\/p><p id=\"fs-id1167833311012\">We know when we solve a system of two linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown.<\/p><span data-type=\"media\" id=\"fs-id1167829598038\" data-alt=\"Figure shows three graphs. In the first one, two lines intersect. Intersecting lines have one point in common. There is one solution to this system. The graph is labeled Consistent Independent. In the second graph, two lines are parallel. Parallel lines have no points in common. There is no solution to this system. The graph is labeled inconsistent. In the third graph, there is just one line. Both equations give the same line. Because we have just one line, there are infinitely many solutions. It is labeled consistent dependent.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows three graphs. In the first one, two lines intersect. Intersecting lines have one point in common. There is one solution to this system. The graph is labeled Consistent Independent. In the second graph, two lines are parallel. Parallel lines have no points in common. There is no solution to this system. The graph is labeled inconsistent. In the third graph, there is just one line. Both equations give the same line. Because we have just one line, there are infinitely many solutions. It is labeled consistent dependent.\"><\/span><p id=\"fs-id1167836608495\">Similarly, for a linear equation with three variables \\(ax+by+cz=d,\\) every solution to the equation is an ordered triple, \\(\\left(x,y,z\\right)\\) that makes the equation true.<\/p><div data-type=\"note\" id=\"fs-id1167836321732\"><div data-type=\"title\">Linear Equation in Three Variables<\/div><p id=\"fs-id1167833129160\">A linear equation with three variables, where <em data-effect=\"italics\">a, b, c,<\/em> and <em data-effect=\"italics\">d<\/em> are real numbers and <em data-effect=\"italics\">a, b<\/em>, and <em data-effect=\"italics\">c<\/em> are not all 0, is of the form<\/p><div data-type=\"equation\" id=\"fs-id1167836601622\" class=\"unnumbered\" data-label=\"\">\\(ax+by+cz=d\\)<\/div><p id=\"fs-id1167833047411\">Every solution to the equation is an ordered triple, \\(\\left(x,y,z\\right)\\) that makes the equation true.<\/p><\/div><p id=\"fs-id1167836502713\">All the points that are solutions to one equation form a plane in three-dimensional space. And, by finding what the planes have in common, we\u2019ll find the solution to the system.<\/p><p id=\"fs-id1167829809276\">When we solve a system of three linear equations represented by a graph of three planes in space, there are three possible cases.<\/p><span data-type=\"media\" id=\"fs-id1167829690706\" data-alt=\"Eight figures are shown. The first one shows three intersecting planes with one point in common. It is labeled Consistent system and Independent equations. The second figure has three parallel planes with no points in common. It is labeled Inconsistent system. In the third figure two planes are coincident and parallel to the third plane. The planes have no points in common. In the fourth figure, two planes are parallel and each intersects the third plane. The planes have no points in common. In the fifth figure, each plane intersects the other two, but all three share no points. The planes have no points in common. In the sixth figure, three planes intersect in one line. There is just one line, so there are infinitely many solutions. In the seventh figure, two planes are coincident and intersect the third plane in a line. There is just one line, so there are infinitely many solutions. In the last figure, three planes are coincident. There is just one plane, so there are infinitely many solutions.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_002h_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Eight figures are shown. The first one shows three intersecting planes with one point in common. It is labeled Consistent system and Independent equations. The second figure has three parallel planes with no points in common. It is labeled Inconsistent system. In the third figure two planes are coincident and parallel to the third plane. The planes have no points in common. In the fourth figure, two planes are parallel and each intersects the third plane. The planes have no points in common. In the fifth figure, each plane intersects the other two, but all three share no points. The planes have no points in common. In the sixth figure, three planes intersect in one line. There is just one line, so there are infinitely many solutions. In the seventh figure, two planes are coincident and intersect the third plane in a line. There is just one line, so there are infinitely many solutions. In the last figure, three planes are coincident. There is just one plane, so there are infinitely many solutions.\"><\/span><span data-type=\"media\" id=\"fs-id1171790367628\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_002a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><span data-type=\"media\" id=\"fs-id1167826129302\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_002b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><span data-type=\"media\" id=\"fs-id1167829830109\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_002c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><span data-type=\"media\" id=\"fs-id1171792823031\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_002d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><span data-type=\"media\" id=\"fs-id1171790368377\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_002e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><span data-type=\"media\" id=\"fs-id1171790448102\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_002f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><span data-type=\"media\" id=\"fs-id1171792480661\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_002g_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><p id=\"fs-id1167836561085\">To solve a system of three linear equations, we want to find the values of the variables that are solutions to all three equations. In other words, we are looking for the ordered triple \\(\\left(x,y,z\\right)\\) that makes all three equations true. These are called the <span data-type=\"term\">solutions of the system of three linear equations with three variables<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1167829752800\"><div data-type=\"title\">Solutions of a System of Linear Equations with Three Variables<\/div><p>Solutions of a system of equations are the values of the variables that make all the equations true. A solution is represented by an <span data-type=\"term\">ordered triple<\/span> \\(\\left(x,y,z\\right).\\)<\/p><\/div><p id=\"fs-id1167836440360\">To determine if an ordered triple is a solution to a system of three equations, we substitute the values of the variables into each equation. If the ordered triple makes all three equations true, it is a solution to the system.<\/p><div data-type=\"example\" id=\"fs-id1167832929828\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829619748\"><div data-type=\"problem\" id=\"fs-id1167836553422\"><p id=\"fs-id1167836333599\">Determine whether the ordered triple is a solution to the system: \\(\\left\\{\\begin{array}{c}x-y+z=2\\hfill \\\\ 2x-y-z=-6\\hfill \\\\ 2x+2y+z=-3\\hfill \\end{array}.\\)<\/p><p id=\"fs-id1167836494346\"><span class=\"token\">\u24d0<\/span>\\(\\left(-2,-1,3\\right)\\)<span class=\"token\">\u24d1<\/span>\\(\\left(-4,-3,4\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829785610\"><p id=\"fs-id1167836392160\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167836512836\" data-alt=\"The equations are x minus y plus z equals 2, 2x minus y minus z equals minus 6 and 2x plus 2y plus z equals minus 3. Substituting minus 2 for x, minus 1 for y and 3 for z into all three equations, we find that all three hold true. Hence, minus 2, minus 1, 3 is a solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_003_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are x minus y plus z equals 2, 2x minus y minus z equals minus 6 and 2x plus 2y plus z equals minus 3. Substituting minus 2 for x, minus 1 for y and 3 for z into all three equations, we find that all three hold true. Hence, minus 2, minus 1, 3 is a solution.\"><\/span><p id=\"fs-id1167833382620\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167825009206\" data-alt=\"The equations are x minus y plus z equals 2, 2x minus y minus z equals minus 6 and 2x plus 2y plus z equals minus 3. Substituting minus minus 4 for x, minus 3 for y and 4 for z into all three equations, we find that all three hold true. Hence, minus 4, minus 3, 4 is not a solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are x minus y plus z equals 2, 2x minus y minus z equals minus 6 and 2x plus 2y plus z equals minus 3. Substituting minus minus 4 for x, minus 3 for y and 4 for z into all three equations, we find that all three hold true. Hence, minus 4, minus 3, 4 is not a solution.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836572810\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836528199\"><p id=\"fs-id1167829744537\">Determine whether the ordered triple is a solution to the system: \\(\\left\\{\\begin{array}{c}3x+y+z=2\\hfill \\\\ x+2y+z=-3\\hfill \\\\ 3x+y+2z=4\\hfill \\end{array}.\\)<\/p><p id=\"fs-id1167829621248\"><span class=\"token\">\u24d0<\/span>\\(\\left(1,-3,2\\right)\\)<span class=\"token\">\u24d1<\/span>\\(\\left(4,-1,-5\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836622967\"><p id=\"fs-id1167822971291\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836628976\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833139268\"><div data-type=\"problem\" id=\"fs-id1167829599455\"><p id=\"fs-id1167829690831\">Determine whether the ordered triple is a solution to the system: \\(\\left\\{\\begin{array}{c}x-3y+z=-5\\hfill \\\\ -3x-y-z=1\\hfill \\\\ 2x-2y+3z=1\\hfill \\end{array}.\\)<\/p><p id=\"fs-id1167836650229\"><span class=\"token\">\u24d0<\/span>\\(\\left(2,-2,3\\right)\\)<span class=\"token\">\u24d1<\/span>\\(\\left(-2,2,3\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829714944\"><p id=\"fs-id1167836546570\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> yes<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836293725\"><h3 data-type=\"title\">Solve a System of Linear Equations with Three Variables<\/h3><p id=\"fs-id1167836287881\">To solve a system of linear equations with three variables, we basically use the same techniques we used with systems that had two variables. We start with two pairs of equations and in each pair we eliminate the same variable. This will then give us a system of equations with only two variables and then we know how to solve that system!<\/p><p id=\"fs-id1167833023923\">Next, we use the values of the two variables we just found to go back to the original equation and find the third variable. We write our answer as an ordered triple and then check our results.<\/p><div data-type=\"example\" id=\"fs-id1167833350276\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Solve a System of Equations With Three Variables by Elimination<\/div><div data-type=\"exercise\" id=\"fs-id1167836729945\"><div data-type=\"problem\"><p id=\"fs-id1167829712854\">Solve the system by elimination: \\(\\left\\{\\begin{array}{c}x-2y+z=3\\hfill \\\\ 2x+y+z=4\\hfill \\\\ 3x+4y+3z=-1\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833369162\"><span data-type=\"media\" id=\"fs-id1167833335338\" data-alt=\"The equations are x minus 2y plus z equals 3, 2x plus y plus z equals 4 and 3x plus 4y plus 3z equals minus 1. Step 1 is to write the equations in standard form. They are. If any coefficients are fractions, clear them. There are none.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_005a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are x minus 2y plus z equals 3, 2x plus y plus z equals 4 and 3x plus 4y plus 3z equals minus 1. Step 1 is to write the equations in standard form. They are. If any coefficients are fractions, clear them. There are none.\"><\/span><span data-type=\"media\" id=\"fs-id1167825830021\" data-alt=\"Step 2 is to eliminate the same variable from two equations. Decide which variable you will eliminate. We can eliminate the y\u2019s from equations 1 and 2 by multiplying equation 2 by 2. Work with a pair of equations to eliminate the chosen variable. Multiply one or both equations so that the coefficients of that variable are opposites. Add the equations resulting from Step 2 to eliminate one variable. The new equation we get is 5x plus 3z equals 11.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_005b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to eliminate the same variable from two equations. Decide which variable you will eliminate. We can eliminate the y\u2019s from equations 1 and 2 by multiplying equation 2 by 2. Work with a pair of equations to eliminate the chosen variable. Multiply one or both equations so that the coefficients of that variable are opposites. Add the equations resulting from Step 2 to eliminate one variable. The new equation we get is 5x plus 3z equals 11.\"><\/span><span data-type=\"media\" data-alt=\"Step 3 is to repeat step 2 using two other equations and eliminate the same variable as in step 2. We can again eliminate the y\u2019s using the equations 1, 3 by multiplying equation 1 by 2. Add the new equations and the result will be 5x plus 5z equals 5.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_005c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to repeat step 2 using two other equations and eliminate the same variable as in step 2. We can again eliminate the y\u2019s using the equations 1, 3 by multiplying equation 1 by 2. Add the new equations and the result will be 5x plus 5z equals 5.\"><\/span><span data-type=\"media\" id=\"fs-id1167836292792\" data-alt=\"Step 4. The two new equations form a system of two equations with two variables. Solve this system. Eliminating x, we get z equal to minus 3. Substituting this in one of the new equations, we get x equal to 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_005d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4. The two new equations form a system of two equations with two variables. Solve this system. Eliminating x, we get z equal to minus 3. Substituting this in one of the new equations, we get x equal to 4.\"><\/span><span data-type=\"media\" id=\"fs-id1167826172504\" data-alt=\"Step 5 is to use the values of the two variables found in step 4 to find the third variable. Substituting values of x and z in one of the original equations, we get y equal to minus 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_005e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to use the values of the two variables found in step 4 to find the third variable. Substituting values of x and z in one of the original equations, we get y equal to minus 1.\"><\/span><span data-type=\"media\" id=\"fs-id1167836611284\" data-alt=\"Step 6 is to write the solution as an ordered triple 4, minus 1, minus 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_005f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 6 is to write the solution as an ordered triple 4, minus 1, minus 3.\"><\/span><span data-type=\"media\" id=\"fs-id1167829790585\" data-alt=\"Step 7 is to check that the ordered triple is a solution to all three original equations. It makes all three equations true.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_005g_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 7 is to check that the ordered triple is a solution to all three original equations. It makes all three equations true.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829752373\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836621335\"><div data-type=\"problem\" id=\"fs-id1167829908545\"><p id=\"fs-id1167830077067\">Solve the system by elimination: \\(\\left\\{\\begin{array}{c}3x+y-z=2\\hfill \\\\ 2x-3y-2z=1\\hfill \\\\ 4x-y-3z=0\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833021838\"><p id=\"fs-id1167822193559\">\\(\\left(2,-1,3\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836295067\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836356451\"><div data-type=\"problem\" id=\"fs-id1167833076732\"><p id=\"fs-id1167822890955\">Solve the system by elimination: \\(\\left\\{\\begin{array}{c}4x+y+z=-1\\hfill \\\\ -2x-2y+z=2\\hfill \\\\ 2x+3y-z=1\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829744332\"><p id=\"fs-id1167826170389\">\\(\\left(-2,3,4\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836389649\">The steps are summarized here.<\/p><div data-type=\"note\" id=\"fs-id1167836665602\" class=\"howto\"><div data-type=\"title\">Solve a system of linear equations with three variables.<\/div><ol id=\"fs-id1167836598867\" type=\"1\" class=\"stepwise\"><li>Write the equations in standard form <ul id=\"fs-id1167836728212\" data-bullet-style=\"bullet\"><li>If any coefficients are fractions, clear them.<\/li><\/ul><\/li><li>Eliminate the same variable from two equations. <ul id=\"fs-id1167836706262\" data-bullet-style=\"bullet\"><li>Decide which variable you will eliminate.<\/li><li>Work with a pair of equations to eliminate the chosen variable.<\/li><li>Multiply one or both equations so that the coefficients of that variable are opposites.<\/li><li>Add the equations resulting from Step 2 to eliminate one variable<\/li><\/ul><\/li><li>Repeat Step 2 using two other equations and eliminate the same variable as in Step 2.<\/li><li>The two new equations form a system of two equations with two variables. Solve this system.<\/li><li>Use the values of the two variables found in Step 4 to find the third variable.<\/li><li>Write the solution as an ordered triple.<\/li><li>Check that the ordered triple is a solution to <strong data-effect=\"bold\">all three<\/strong> original equations.<\/li><\/ol><\/div><div data-type=\"example\" id=\"fs-id1167833050653\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167832936787\"><div data-type=\"problem\" id=\"fs-id1167833020621\"><p id=\"fs-id1167833381373\">Solve: \\(\\left\\{\\begin{array}{c}3x-4z=0\\hfill \\\\ 3y+2z=-3\\hfill \\\\ 2x+3y=-5\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829593667\"><div data-type=\"equation\" id=\"fs-id1171784163338\" class=\"unnumbered\" data-label=\"\">\\(\\left\\{\\begin{array}{c}3x-4z=0\\phantom{\\rule{1em}{0ex}}\\left(1\\right)\\hfill \\\\ 3y+2z=-3\\phantom{\\rule{0.3em}{0ex}}\\left(2\\right)\\hfill \\\\ 2x+3y=-5\\phantom{\\rule{0.3em}{0ex}}\\left(3\\right)\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167836611848\">We can eliminate \\(z\\) from equations (1) and (2) by multiplying equation (2) by 2 and then adding the resulting equations.<\/p><span data-type=\"media\" id=\"fs-id1167826206324\" data-alt=\"The equations are 3 x minus 4 equals 0, 3y plus 2 z equals minus 3 and 2 x plus 3 y equals minus 5. Multiply equation 2 by 2 and add to equation 1. We get 3 x plus 6 y equals minus 6.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_006a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are 3 x minus 4 equals 0, 3y plus 2 z equals minus 3 and 2 x plus 3 y equals minus 5. Multiply equation 2 by 2 and add to equation 1. We get 3 x plus 6 y equals minus 6.\"><\/span><p id=\"fs-id1167829784503\">Notice that equations (3) and (4) both have the variables \\(x\\) and \\(y\\). We will solve this new system for \\(x\\) and \\(y\\).<\/p><span data-type=\"media\" id=\"fs-id1167829906127\" data-alt=\"Multiply equation 3 by minus 2 and add that to equation 4. We get x equal to minus 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_006b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Multiply equation 3 by minus 2 and add that to equation 4. We get x equal to minus 4.\"><\/span><p id=\"fs-id1167829810622\">To solve for <em data-effect=\"italics\">y<\/em>, we substitute \\(x=-4\\) into equation (3).<\/p><span data-type=\"media\" id=\"fs-id1167833369395\" data-alt=\"Substitute minus 4 into equation 3 and solve for y. We get y equal to 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_006c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Substitute minus 4 into equation 3 and solve for y. We get y equal to 1.\"><\/span><p id=\"fs-id1167836714302\">We now have \\(x=-4\\) and \\(y=1.\\) We need to solve for <em data-effect=\"italics\">z<\/em>. We can substitute \\(x=-4\\) into equation (1) to find <em data-effect=\"italics\">z<\/em>.<\/p><span data-type=\"media\" data-alt=\"Substituting minus 4 into equation 1 for x, we get z equal to minus 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_006d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Substituting minus 4 into equation 1 for x, we get z equal to minus 3.\"><\/span><p id=\"fs-id1167832978282\">We write the solution as an ordered triple. \\(\\phantom{\\rule{6em}{0ex}}\\left(-4,1,-3\\right)\\)<\/p><p id=\"fs-id1167836573935\">We check that the solution makes all three equations true.<\/p><p>\\(\\begin{array}{ccccccc}\\begin{array}{ccc}\\hfill 3x-4z&amp; =\\hfill &amp; 0\\left(1\\right)\\hfill \\\\ \\hfill 3\\left(-4\\right)-4\\left(-3\\right)&amp; \\stackrel{?}{=}\\hfill &amp; 0\\hfill \\\\ \\hfill 0&amp; =\\hfill &amp; 0\u2713\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{ccc}\\hfill 3y+2z&amp; =\\hfill &amp; -3\\left(2\\right)\\hfill \\\\ \\hfill 3\\left(1\\right)+2\\left(-3\\right)&amp; \\stackrel{?}{=}\\hfill &amp; -3\\hfill \\\\ \\hfill -3&amp; =\\hfill &amp; -3\u2713\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{}\\\\ \\\\ \\begin{array}{ccc}\\hfill 2x+3y&amp; =\\hfill &amp; -5\\left(3\\right)\\hfill \\\\ \\hfill 2\\left(-4\\right)+3\\left(1\\right)&amp; \\stackrel{?}{=}\\hfill &amp; -5\\hfill \\\\ \\hfill -5&amp; =\\hfill &amp; -5\u2713\\hfill \\end{array}\\hfill \\\\ \\text{The solution is}\\phantom{\\rule{0.2em}{0ex}}\\left(-4,1,-3\\right).\\hfill \\end{array}\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832940114\"><div data-type=\"problem\"><p id=\"fs-id1167833382031\">Solve: \\(\\left\\{\\begin{array}{c}3x-4z=-1\\hfill \\\\ 2y+3z=2\\hfill \\\\ 2x+3y=6\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836704561\"><p id=\"fs-id1167836620807\">\\(\\left(-3,4,-2\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167825782333\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836627315\"><div data-type=\"problem\" id=\"fs-id1167833382121\"><p id=\"fs-id1167833138643\">Solve: \\(\\left\\{\\begin{array}{c}4x-3z=-5\\hfill \\\\ 3y+2z=7\\hfill \\\\ 3x+4y=6\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836434358\"><p id=\"fs-id1167836522895\">\\(\\left(-2,3,-1\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167833138169\">When we solve a system and end up with no variables and a false statement, we know there are no solutions and that the system is inconsistent. The next example shows a system of equations that is inconsistent.<\/p><div data-type=\"example\" id=\"fs-id1167829704831\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836576093\"><div data-type=\"problem\"><p id=\"fs-id1167829878516\">Solve the system of equations: \\(\\left\\{\\begin{array}{c}x+2y-3z=-1\\hfill \\\\ x-3y+z=1\\hfill \\\\ 2x-y-2z=2\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836624752\"><div data-type=\"equation\" id=\"fs-id1171791597904\" class=\"unnumbered\" data-label=\"\">\\(\\left\\{\\begin{array}{c}x+2y-3z=-1\\phantom{\\rule{0.3em}{0ex}}\\left(1\\right)\\hfill \\\\ x-3y+z=1\\phantom{\\rule{1.5em}{0ex}}\\left(2\\right)\\hfill \\\\ 2x-y-2z=2\\phantom{\\rule{1em}{0ex}}\\left(3\\right)\\hfill \\end{array}\\)<\/div><p>Use equation (1) and (2) to eliminate <em data-effect=\"italics\">z<\/em>.<\/p><span data-type=\"media\" id=\"fs-id1167833021780\" data-alt=\"The equations are x plus 2y minus 3z equals minus 1, x minus 3y plus z equals 1 and 2x minus y minus 2z equals 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_007a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are x plus 2y minus 3z equals minus 1, x minus 3y plus z equals 1 and 2x minus y minus 2z equals 2.\"><\/span><p>Use (2) and (3) to eliminate \\(z\\) again.<\/p><span data-type=\"media\" id=\"fs-id1167829719422\" data-alt=\"Multiplying equation 2 by 3 and adding it to equation 1, we get equation 4, 4x minus 7y equals 2. Multiplying equation 2 by 2 and adding it to equation 3, we get equation 5, 4x minus 7y equals 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_007b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Multiplying equation 2 by 3 and adding it to equation 1, we get equation 4, 4x minus 7y equals 2. Multiplying equation 2 by 2 and adding it to equation 3, we get equation 5, 4x minus 7y equals 4.\"><\/span><p id=\"fs-id1167836521885\">Use (4) and (5) to eliminate a variable.<\/p><span data-type=\"media\" data-alt=\"Equations 4 and 5 both have 2 variables. Multiply equation 5 by minus 1 and add it to equation 4. We get 0 equal to minus 2, which is false.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_007c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Equations 4 and 5 both have 2 variables. Multiply equation 5 by minus 1 and add it to equation 4. We get 0 equal to minus 2, which is false.\"><\/span><p id=\"fs-id1167824731658\">There is no solution.<\/p><p>We are left with a false statement and this tells us the system is inconsistent and has no solution.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836407993\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836334898\"><div data-type=\"problem\"><p id=\"fs-id1167836755457\">Solve the system of equations: \\(\\left\\{\\begin{array}{c}x+2y+6z=5\\hfill \\\\ -x+y-2z=3\\hfill \\\\ x-4y-2z=1\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836788456\"><p id=\"fs-id1167829984082\">no solution<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829791296\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167833057146\"><p>Solve the system of equations: \\(\\left\\{\\begin{array}{c}2x-2y+3z=6\\hfill \\\\ 4x-3y+2z=0\\hfill \\\\ -2x+3y-7z=1\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836574853\"><p id=\"fs-id1167836456023\">no solution<\/p><\/div><\/div><\/div><p id=\"fs-id1167830004605\">When we solve a system and end up with no variables but a true statement, we know there are infinitely many solutions. The system is consistent with dependent equations. Our solution will show how two of the variables depend on the third.<\/p><div data-type=\"example\" id=\"fs-id1167836510584\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836398862\"><div data-type=\"problem\"><p id=\"fs-id1167836534914\">Solve the system of equations: \\(\\left\\{\\begin{array}{c}x+2y-z=1\\hfill \\\\ 2x+7y+4z=11\\hfill \\\\ x+3y+z=4\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836286713\"><div data-type=\"equation\" id=\"fs-id1171791735385\" class=\"unnumbered\" data-label=\"\">\\(\\left\\{\\begin{array}{c}x+2y-z=1\\phantom{\\rule{1.7em}{0ex}}\\left(1\\right)\\hfill \\\\ 2x+7y+4z=11\\phantom{\\rule{0.3em}{0ex}}\\left(2\\right)\\hfill \\\\ x+3y+z=4\\phantom{\\rule{1.7em}{0ex}}\\left(3\\right)\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167833139042\">Use equation (1) and (3) to eliminate <em data-effect=\"italics\">x<\/em>.<\/p><span data-type=\"media\" id=\"fs-id1167836508275\" data-alt=\"The equations are x plus 2y minus z equals 1, 2x plus 7y plus 4z equals 11 and x plus 3y plus z equals 4. Multiply equation 1 with minus 1 and add it to equation 3. We get equation 4, y plus 2z equals 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_008a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are x plus 2y minus z equals 1, 2x plus 7y plus 4z equals 11 and x plus 3y plus z equals 4. Multiply equation 1 with minus 1 and add it to equation 3. We get equation 4, y plus 2z equals 3.\"><\/span><p id=\"fs-id1167833022192\">Use equation (1) and (2) to eliminate <em data-effect=\"italics\">x<\/em> again.<\/p><span data-type=\"media\" id=\"fs-id1167826171290\" data-alt=\"Multiply equation 1 with minus 2 and add it to equation 2. We get equation 5, 3y plus 6z equals 9.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_008b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Multiply equation 1 with minus 2 and add it to equation 2. We get equation 5, 3y plus 6z equals 9.\"><\/span><p id=\"fs-id1167836327517\">Use equation (4) and (5) to eliminate \\(y\\).<\/p><span data-type=\"media\" id=\"fs-id1171789696361\" data-alt=\"Multiply equation 4 with minus 3 and add it to equation 5. We get 0 equal to 0. There are infinite many solutions. Solving equation 4 for y, we get y equal to minus 2z plus 3. Substituting this into equation 1, we get x equal to 5z minus 5. The true statement 0 equal to 0 tells us that this is a dependent system that has infinitely many solutions. The solutions are of the form x, y, z where x is 5z minus 5, y is minus 2z plus 3 and z is any real number.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_008c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Multiply equation 4 with minus 3 and add it to equation 5. We get 0 equal to 0. There are infinite many solutions. Solving equation 4 for y, we get y equal to minus 2z plus 3. Substituting this into equation 1, we get x equal to 5z minus 5. The true statement 0 equal to 0 tells us that this is a dependent system that has infinitely many solutions. The solutions are of the form x, y, z where x is 5z minus 5, y is minus 2z plus 3 and z is any real number.\"><\/span><table id=\"fs-id1167824774013\" class=\"unnumbered unstyled\" summary=\"\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">There are infinitely many solutions.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve equation (4) for <em data-effect=\"italics\">y<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\">Represent the solution showing how <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> are dependent on <em data-effect=\"italics\">z<\/em>.<div data-type=\"newline\"><br><\/div>\\(\\begin{array}{ccc}\\hfill y+2z&amp; =\\hfill &amp; 3\\hfill \\\\ \\hfill y&amp; =\\hfill &amp; -2z+3\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use equation (1) to solve for <em data-effect=\"italics\">x<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{4em}{0ex}}x+2y-z=1\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute \\(y=-2z+3.\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\begin{array}{ccc}\\hfill x+2\\left(-2z+3\\right)-z&amp; =\\hfill &amp; 1\\hfill \\\\ \\hfill x-4z+6-z&amp; =\\hfill &amp; 1\\hfill \\\\ \\hfill x-5z+6&amp; =\\hfill &amp; 1\\hfill \\\\ \\hfill x&amp; =\\hfill &amp; 5z-5\\hfill \\end{array}\\)<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836360117\">The true statement \\(0=0\\) tells us that this is a dependent system that has infinitely many solutions. The solutions are of the form \\(\\left(x,y,z\\right)\\) where \\(x=5z-5;y=-2z+3\\)and <em data-effect=\"italics\">z<\/em> is any real number.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829907255\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829796752\"><div data-type=\"problem\" id=\"fs-id1167829718219\"><p id=\"fs-id1167836492090\">Solve the system by equations: \\(\\left\\{\\begin{array}{c}x+y-z=0\\hfill \\\\ 2x+4y-2z=6\\hfill \\\\ 3x+6y-3z=9\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829861781\"><p id=\"fs-id1167836531064\">infinitely many solutions\\(\\left(x,3,z\\right)\\) where \\(x=z-3;y=3;z\\) is any real number<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836530163\"><div data-type=\"problem\" id=\"fs-id1167836339987\"><p id=\"fs-id1167829719789\">Solve the system by equations: \\(\\left\\{\\begin{array}{c}x-y-z=1\\hfill \\\\ -x+2y-3z=-4\\hfill \\\\ 3x-2y-7z=0\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836666754\"><p id=\"fs-id1167833086865\">infinitely many solutions \\(\\left(x,y,z\\right)\\) where\\(x=5z-2;y=4z-3;z\\) is any real number<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836539512\"><h3 data-type=\"title\">Solve Applications using Systems of Linear Equations with Three Variables<\/h3><p id=\"fs-id1167833086461\">Applications that are modeled by a systems of equations can be solved using the same techniques we used to solve the systems. Many of the application are just extensions to three variables of the types we have solved earlier.<\/p><div data-type=\"example\" id=\"fs-id1167836622870\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167833227340\"><div data-type=\"problem\" id=\"fs-id1167829624534\"><p>The community college theater department sold three kinds of tickets to its latest play production. The adult tickets sold for ?15, the student tickets for ?10 and the child tickets for ?8. The theater department was thrilled to have sold 250 tickets and brought in ?2,825 in one night. The number of student tickets sold is twice the number of adult tickets sold. How many of each type did the department sell?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829693376\"><table id=\"fs-id1167829812971\" class=\"unnumbered unstyled can-break\" summary=\"We will use a chart to organize information. The type is adult, student and child. The numbers for these are x, y and z respectively and the total is 250. The values for these are 15, 10 and 8 respectively. The total values are 15x, 10y and 8z respectively, the total being 2825. Number of students is twice number of adults. So y is 2x. We rewrite as 2x minus y equals 0. The system of equations is x plus y plus z equals 250, 15x plus 10y plus 8z is 2825 and minus 2x plus y is 0. Multiply equation 1 by minus 8 and add it to equation 2. We get 7x plus 2y equals 825. Multiply equation 3 with minus 2 and add it to equation 4. Solving for x, we get x equal to 75. Substituting this into equation 3, we get y equal to 150. Substituting values of x and y in equation 1, we get z equal to 25. The theater department sold 75 adult tickets, 150 student tickets, and 25 child tickets.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">We will use a chart to organize the information.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836693208\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Number of students is twice number of adults.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"bottom\" data-align=\"left\">Rewrite the equation in standard form.<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\begin{array}{ccc}\\hfill y&amp; =\\hfill &amp; 2x\\hfill \\\\ \\hfill 2x-y&amp; =\\hfill &amp; 0\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836449342\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_009b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use equations (1) and (2) to eliminate <em data-effect=\"italics\">z<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_009c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use (3) and (4) to eliminate \\(y.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836545039\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_009d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">x<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{1.3em}{0ex}}\u2003x\\phantom{\\rule{3.6em}{0ex}}=75\\) adult tickets<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use equation (3) to find <em data-effect=\"italics\">y<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{3.4em}{0ex}}-2x+y=0\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute \\(x=75.\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{1.7em}{0ex}}\\begin{array}{ccc}\\hfill -2\\left(75\\right)+y&amp; =\\hfill &amp; 0\\hfill \\\\ \\hfill -150+y&amp; =\\hfill &amp; 0\\hfill \\\\ \\hfill y&amp; =\\hfill &amp; 150\\phantom{\\rule{0.2em}{0ex}}\\text{student tickets}\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use equation (1) to find <em data-effect=\"italics\">z<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{3.1em}{0ex}}x+y+z=250\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute in the values<div data-type=\"newline\"><br><\/div>\\(x=75,\\phantom{\\rule{0.2em}{0ex}}y=150.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{1em}{0ex}}\\begin{array}{ccc}\\hfill 75+150+z&amp; =\\hfill &amp; 250\\hfill \\\\ \\hfill 225+z&amp; =\\hfill &amp; 250\\hfill \\\\ \\hfill z&amp; =\\hfill &amp; 25\\phantom{\\rule{0.2em}{0ex}}\\text{child tickets}\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the solution.<\/td><td data-valign=\"top\" data-align=\"left\">The theater department sold 75 adult tickets,<div data-type=\"newline\"><br><\/div>150 student tickets, and 25 child tickets.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833279849\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829651324\"><div data-type=\"problem\" id=\"fs-id1167836449776\"><p id=\"fs-id1167832976579\">The community college fine arts department sold three kinds of tickets to its latest dance presentation. The adult tickets sold for ?20, the student tickets for ?12 and the child tickets for ?10.The fine arts department was thrilled to have sold 350 tickets and brought in ?4,650 in one night. The number of child tickets sold is the same as the number of adult tickets sold. How many of each type did the department sell?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829811278\"><p id=\"fs-id1167829715922\">The fine arts department sold 75 adult tickets, 200 student tickets, and 75 child tickets.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836330695\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167829904743\"><p>The community college soccer team sold three kinds of tickets to its latest game. The adult tickets sold for ?10, the student tickets for ?8 and the child tickets for ?5. The soccer team was thrilled to have sold 600 tickets and brought in ?4,900 for one game. The number of adult tickets is twice the number of child tickets. How many of each type did the soccer team sell?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836627190\"><p id=\"fs-id1167825872620\">The soccer team sold 200 adult tickets, 300 student tickets, and 100 child tickets.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836509791\" class=\"media-2\"><p id=\"fs-id1167829831129\">Access this online resource for additional instruction and practice with solving a linear system in three variables with no or infinite solutions.<\/p><ul id=\"fs-id1169146669576\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37linsys3var\">Solving a Linear System in Three Variables with No or Infinite Solutions<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37variableapp\">3 Variable Application<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167836774761\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Linear Equation in Three Variables:<\/strong> A linear equation with three variables, where <em data-effect=\"italics\">a, b, c<\/em>, and <em data-effect=\"italics\">d<\/em> are real numbers and <em data-effect=\"italics\">a, b,<\/em> and <em data-effect=\"italics\">c<\/em> are not all 0, is of the form<div data-type=\"newline\"><br><\/div> <div data-type=\"equation\" id=\"fs-id1167829743566\" class=\"unnumbered\" data-label=\"\">\\(ax+by+cz=d\\)<\/div><div data-type=\"newline\"><br><\/div> Every solution to the equation is an ordered triple, \\(\\left(x,y,z\\right)\\) that makes the equation true.<\/li><li><strong data-effect=\"bold\">How to solve a system of linear equations with three variables.<\/strong><ol id=\"fs-id1167833365883\" type=\"1\" class=\"stepwise\"><li>Write the equations in standard form<div data-type=\"newline\"><br><\/div> If any coefficients are fractions, clear them.<\/li><li>Eliminate the same variable from two equations.<div data-type=\"newline\"><br><\/div> Decide which variable you will eliminate.<div data-type=\"newline\"><br><\/div> Work with a pair of equations to eliminate the chosen variable.<div data-type=\"newline\"><br><\/div> Multiply one or both equations so that the coefficients of that variable are opposites.<div data-type=\"newline\"><br><\/div> Add the equations resulting from Step 2 to eliminate one variable<\/li><li>Repeat Step 2 using two other equations and eliminate the same variable as in Step 2.<\/li><li>The two new equations form a system of two equations with two variables. Solve this system.<\/li><li>Use the values of the two variables found in Step 4 to find the third variable.<\/li><li>Write the solution as an ordered triple.<\/li><li>Check that the ordered triple is a solution to <strong data-effect=\"bold\">all three<\/strong> original equations.<\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167829906194\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167829634077\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167830096283\"><strong data-effect=\"bold\">Determine Whether an Ordered Triple is a Solution of a System of Three Linear Equations with Three Variables<\/strong><\/p><p id=\"fs-id1167829936718\">In the following exercises, determine whether the ordered triple is a solution to the system.<\/p><div data-type=\"exercise\" id=\"fs-id1167836501642\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833086359\"><p>\\(\\left\\{\\begin{array}{c}2x-6y+z=3\\hfill \\\\ 3x-4y-3z=2\\hfill \\\\ 2x+3y-2z=3\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167836706047\"><span class=\"token\">\u24d0<\/span>\\(\\left(3,1,3\\right)\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(4,3,7\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836510397\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167824748848\"><p id=\"fs-id1167829651306\">\\(\\left\\{\\begin{array}{c}-3x+\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}y+z=-4\\hfill \\\\ -x+2y-2z=1\\hfill \\\\ 2x-\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}y-z=-1\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167836287729\"><span class=\"token\">\u24d0<\/span>\\(\\left(-5,-7,4\\right)\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(5,7,4\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167836573198\"><p id=\"fs-id1167829807906\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> yes<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833136684\" class=\"material-set-2\"><div data-type=\"problem\"><p id=\"fs-id1167833021377\">\\(\\left\\{\\begin{array}{c}y-10z=-8\\hfill \\\\ 2x-y=2\\hfill \\\\ x-5z=3\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167833046978\"><span class=\"token\">\u24d0<\/span>\\(\\left(7,12,2\\right)\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(2,2,1\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836686650\" class=\"material-set-2\"><div data-type=\"problem\"><p id=\"fs-id1167836686455\">\\(\\left\\{\\begin{array}{c}x+3y-z=15\\hfill \\\\ y=\\frac{2}{3}x-2\\hfill \\\\ x-3y+z=-2\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167829741854\"><span class=\"token\">\u24d0<\/span>\\(\\left(-6,5,\\frac{1}{2}\\right)\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(5,\\frac{4}{3},-3\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167829696536\"><p id=\"fs-id1167829598072\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> yes<\/p><\/div><\/div><p id=\"fs-id1167829713554\"><strong data-effect=\"bold\">Solve a System of Linear Equations with Three Variables<\/strong><\/p><p>In the following exercises, solve the system of equations.<\/p><div data-type=\"exercise\" id=\"fs-id1167833327314\"><div data-type=\"problem\" id=\"fs-id1167836390323\"><p id=\"fs-id1167836738284\">\\(\\left\\{\\begin{array}{c}5x+2y+z=5\\hfill \\\\ -3x-y+2z=6\\hfill \\\\ 2x+3y-3z=5\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833086748\"><div data-type=\"problem\" id=\"fs-id1167824740545\"><p id=\"fs-id1167829872202\">\\(\\left\\{\\begin{array}{c}6x-5y+2z=3\\hfill \\\\ 2x+y-4z=5\\hfill \\\\ 3x-3y+z=-1\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829749058\"><p id=\"fs-id1167836516626\">\\(\\left(4,5,2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836392614\"><div data-type=\"problem\" id=\"fs-id1167824774411\"><p id=\"fs-id1167829594499\">\\(\\left\\{\\begin{array}{c}2x-5y+3z=8\\hfill \\\\ 3x-y+4z=7\\hfill \\\\ x+3y+2z=-3\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836493253\"><div data-type=\"problem\" id=\"fs-id1167836288151\"><p id=\"fs-id1167829717239\">\\(\\left\\{\\begin{array}{c}5x-3y+2z=-5\\hfill \\\\ 2x-y-z=4\\hfill \\\\ 3x-2y+2z=-7\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824764689\"><p id=\"fs-id1167836703915\">\\(\\left(7,12,-2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836477516\"><div data-type=\"problem\" id=\"fs-id1167836508453\"><p>\\(\\left\\{\\begin{array}{c}3x-5y+4z=5\\hfill \\\\ 5x+2y+z=0\\hfill \\\\ 2x+3y-2z=3\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829878819\"><div data-type=\"problem\" id=\"fs-id1167836547728\"><p id=\"fs-id1167836691963\">\\(\\left\\{\\begin{array}{c}4x-3y+z=7\\hfill \\\\ 2x-5y-4z=3\\hfill \\\\ 3x-2y-2z=-7\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826170426\"><p id=\"fs-id1167829930733\">\\(\\left(-3,-5,4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830013969\"><div data-type=\"problem\" id=\"fs-id1167836481012\"><p id=\"fs-id1167833025488\">\\(\\left\\{\\begin{array}{c}3x+8y+2z=-5\\hfill \\\\ 2x+5y-3z=0\\hfill \\\\ x+2y-2z=-1\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832999524\"><div data-type=\"problem\" id=\"fs-id1167836532597\"><p id=\"fs-id1167829716811\">\\(\\left\\{\\begin{array}{c}11x+9y+2z=-9\\hfill \\\\ 7x+5y+3z=-7\\hfill \\\\ 4x+3y+z=-3\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832925579\"><p id=\"fs-id1167829786231\">\\(\\left(2,-3,-2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836689151\"><div data-type=\"problem\" id=\"fs-id1167824774450\"><p id=\"fs-id1167836515372\">\\(\\left\\{\\begin{array}{c}\\frac{1}{3}x-y-z=1\\hfill \\\\ x+\\frac{5}{2}y+z=-2\\hfill \\\\ 2x+2y+\\frac{1}{2}z=-4\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836614091\"><div data-type=\"problem\" id=\"fs-id1167829711744\"><p id=\"fs-id1167836705310\">\\(\\left\\{\\begin{array}{c}x+\\frac{1}{2}y+\\frac{1}{2}z=0\\hfill \\\\ \\frac{1}{5}x-\\frac{1}{5}y+z=0\\hfill \\\\ \\frac{1}{3}x-\\frac{1}{3}y+2z=-1\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836300586\"><p id=\"fs-id1167836415047\">\\(\\left(6,-9,-3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836598004\"><div data-type=\"problem\" id=\"fs-id1167833272219\"><p id=\"fs-id1167830077526\">\\(\\left\\{\\begin{array}{c}x+\\frac{1}{3}y-2z=-1\\hfill \\\\ \\frac{1}{3}x+y+\\frac{1}{2}z=0\\hfill \\\\ \\frac{1}{2}x+\\frac{1}{3}y-\\frac{1}{2}z=-1\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836510914\"><div data-type=\"problem\" id=\"fs-id1167836732591\"><p id=\"fs-id1167829787685\">\\(\\left\\{\\begin{array}{c}\\frac{1}{3}x-y+\\frac{1}{2}z=4\\hfill \\\\ \\frac{2}{3}x+\\frac{5}{2}y-4z=0\\hfill \\\\ x-\\frac{1}{2}y+\\frac{3}{2}z=2\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829745800\"><p id=\"fs-id1167836597056\">\\(\\left(3,-4,-2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836362489\"><div data-type=\"problem\" id=\"fs-id1167836552028\"><p id=\"fs-id1167836585257\">\\(\\left\\{\\begin{array}{c}x+2z=0\\hfill \\\\ 4y+3z=-2\\hfill \\\\ 2x-5y=3\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829748029\"><div data-type=\"problem\" id=\"fs-id1167829905146\"><p id=\"fs-id1167836508175\">\\(\\left\\{\\begin{array}{c}2x+5y=4\\hfill \\\\ 3y-z=3\\hfill \\\\ 4x+3z=-3\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829580059\"><p id=\"fs-id1167836549784\">\\(\\left(-3,2,3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829807670\"><div data-type=\"problem\" id=\"fs-id1167833138383\"><p id=\"fs-id1167829686513\">\\(\\left\\{\\begin{array}{c}2y+3z=-1\\hfill \\\\ 5x+3y=-6\\hfill \\\\ 7x+z=1\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829740446\"><div data-type=\"problem\" id=\"fs-id1167833329016\"><p id=\"fs-id1167829989579\">\\(\\left\\{\\begin{array}{c}3x-z=-3\\hfill \\\\ 5y+2z=-6\\hfill \\\\ 4x+3y=-8\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836542169\"><p id=\"fs-id1167836387902\">\\(\\left(-2,0,-3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836532846\"><div data-type=\"problem\" id=\"fs-id1167829744480\"><p id=\"fs-id1167836545984\">\\(\\left\\{\\begin{array}{c}4x-3y+2z=0\\hfill \\\\ -2x+3y-7z=1\\hfill \\\\ 2x-2y+3z=6\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829811216\"><div data-type=\"problem\" id=\"fs-id1167836447777\"><p id=\"fs-id1167836299522\">\\(\\left\\{\\begin{array}{c}x-2y+2z=1\\hfill \\\\ -2x+y-z=2\\hfill \\\\ x-y+z=5\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836664947\"><p id=\"fs-id1167833397142\">no solution<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829850954\"><div data-type=\"problem\" id=\"fs-id1167829844151\"><p id=\"fs-id1167833009058\">\\(\\left\\{\\begin{array}{c}2x+3y+z=12\\hfill \\\\ x+y+z=9\\hfill \\\\ 3x+4y+2z=20\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836524317\"><div data-type=\"problem\" id=\"fs-id1167829893475\"><p id=\"fs-id1167825791285\">\\(\\left\\{\\begin{array}{c}x+4y+z=-8\\hfill \\\\ 4x-y+3z=9\\hfill \\\\ 2x+7y+z=0\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833240113\"><p id=\"fs-id1167836507837\">\\(x=\\frac{203}{16};y=\\frac{\u201325}{16};z=\\frac{\u2013231}{16};\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829930558\"><div data-type=\"problem\" id=\"fs-id1167836292764\"><p id=\"fs-id1167836635633\">\\(\\left\\{\\begin{array}{c}x+2y+z=4\\hfill \\\\ x+y-2z=3\\hfill \\\\ -2x-3y+z=-7\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836318721\"><div data-type=\"problem\" id=\"fs-id1167836318723\"><p id=\"fs-id1167825836140\">\\(\\left\\{\\begin{array}{c}x+y-2z=3\\hfill \\\\ -2x-3y+z=-7\\hfill \\\\ x+2y+z=4\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829593597\"><p id=\"fs-id1167829593599\">\\(\\left(x,y,z\\right)\\) where \\(x=5z+2;y=-3z+1;z\\) is any real number<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833135136\"><div data-type=\"problem\" id=\"fs-id1167832961255\"><p id=\"fs-id1167832961257\">\\(\\left\\{\\begin{array}{c}x+y-3z=-1\\hfill \\\\ y-z=0\\hfill \\\\ -x+2y=1\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829598896\"><div data-type=\"problem\" id=\"fs-id1167836693792\"><p id=\"fs-id1167836693795\">\\(\\left\\{\\begin{array}{c}x-2y+3z=1\\hfill \\\\ x+y-3z=7\\hfill \\\\ 3x-4y+5z=7\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833290526\"><p id=\"fs-id1167833290529\">\\(\\left(x,y,z\\right)\\) where \\(x=5z-2;y=4z-3;z\\) is any real number<\/p><\/div><\/div><p id=\"fs-id1167836448592\"><strong data-effect=\"bold\">Solve Applications using Systems of Linear Equations with Three Variables<\/strong><\/p><p id=\"fs-id1167836501809\">In the following exercises, solve the given problem.<\/p><div data-type=\"exercise\" id=\"fs-id1167836501812\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836501814\"><p id=\"fs-id1167833114798\">The sum of the measures of the angles of a triangle is 180. The sum of the measures of the second and third angles is twice the measure of the first angle. The third angle is twelve more than the second. Find the measures of the three angles.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836561270\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836561272\"><p id=\"fs-id1167829712734\">The sum of the measures of the angles of a triangle is 180. The sum of the measures of the second and third angles is three times the measure of the first angle. The third angle is fifteen more than the second. Find the measures of the three angles.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836625011\"><p id=\"fs-id1167836625013\">42, 50, 58<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836556299\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836556301\"><p id=\"fs-id1167836556303\">After watching a major musical production at the theater, the patrons can purchase souvenirs. If a family purchases 4 t-shirts, the video, and 1 stuffed animal, their total is ?135.<\/p><p id=\"fs-id1167836618962\">A couple buys 2 t-shirts, the video, and 3 stuffed animals for their nieces and spends ?115. Another couple buys 2 t-shirts, the video, and 1 stuffed animal and their total is ?85. What is the cost of each item?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836525390\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167830077435\"><p id=\"fs-id1167830077437\">The church youth group is selling snacks to raise money to attend their convention. Amy sold 2 pounds of candy, 3 boxes of cookies and 1 can of popcorn for a total sales of ?65. Brian sold 4 pounds of candy, 6 boxes of cookies and 3 cans of popcorn for a total sales of ?140. Paulina sold 8 pounds of candy, 8 boxes of cookies and 5 cans of popcorn for a total sales of ?250. What is the cost of each item?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830077439\"><p id=\"fs-id1167836694284\">?20, ?5, ?10<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167829905105\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167829807546\"><div data-type=\"problem\" id=\"fs-id1167829807548\"><p>In your own words explain the steps to solve a system of linear equations with three variables by elimination.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833047500\"><div data-type=\"problem\" id=\"fs-id1167836518239\"><p id=\"fs-id1167836518241\">How can you tell when a system of three linear equations with three variables has no solution? Infinitely many solutions?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836578708\"><p id=\"fs-id1167836578710\">Answers will vary.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836387211\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167836477611\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p><span data-type=\"media\" id=\"fs-id1167836705926\" data-alt=\"This table has 4 columns, 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don\u2019t get it. The first row contains the following statements: determine whether an ordered triple is a solution of a system of three linear equations with three variables, solve a system of linear equations with three variables, solve applications using systems of linear equations with three variables. The remaining columns are blank.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns, 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don\u2019t get it. The first row contains the following statements: determine whether an ordered triple is a solution of a system of three linear equations with three variables, solve a system of linear equations with three variables, solve applications using systems of linear equations with three variables. The remaining columns are blank.\"><\/span><p id=\"fs-id1167829596900\"><span class=\"token\">\u24d1<\/span> On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1167829719698\"><dt>solutions of a system of linear equations with three variables<\/dt><dd id=\"fs-id1167825830106\">The solutions of a system of equations are the values of the variables that make all the equations true; a solution is represented by an ordered triple \\(\\left(x,y,z\\right).\\)<\/dd><\/dl><\/div>\n","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to: <\/p>\n<ul>\n<li>Determine whether an ordered triple is a solution of a system of three linear equations with three variables<\/li>\n<li>Solve a system of linear equations with three variables<\/li>\n<li>Solve applications using systems of linear equations with three variables<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829709277\" class=\"be-prepared\">\n<p id=\"fs-id1167833311246\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167833055966\" type=\"1\">\n<li>Evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-203e9ce5d8051a5abc890e122c5e6066_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;&#45;&#50;&#121;&#43;&#51;&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"99\" style=\"vertical-align: -4px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ffba38287436639a3011d50b97654cd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"61\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d5cc0600fefd9fdcd926f46d9373f93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1fc1b68f7e9634bf706f258aabf8d1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"46\" style=\"vertical-align: 0px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167832053133\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Classify the equations as a conditional equation, an identity, or a contradiction and then state the solution. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-86373ea0b7246f8750b1eb4d0e48edff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#50;&#120;&#43;&#121;&#61;&#45;&#49;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#51;&#121;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"150\" style=\"vertical-align: -17px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167836666645\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Classify the equations as a conditional equation, an identity, or a contradiction and then state the solution. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d26f084254f633bd04d51b68d993b877_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#55;&#120;&#43;&#56;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#53;&#121;&#61;&#50;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"132\" style=\"vertical-align: -17px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167826211749\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836729743\">\n<h3 data-type=\"title\">Determine Whether an Ordered Triple is a Solution of a System of Three Linear Equations with Three Variables<\/h3>\n<p id=\"fs-id1167836543656\">In this section, we will extend our work of solving a system of linear equations. So far we have worked with <span data-type=\"term\" class=\"no-emphasis\">systems of equations<\/span> with two equations and two variables. Now we will work with systems of three equations with three variables. But first let&#8217;s review what we already know about solving equations and systems involving up to two variables.<\/p>\n<p id=\"fs-id1167836542288\">We learned earlier that the graph of a <span data-type=\"term\" class=\"no-emphasis\">linear equation<\/span>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc8fab6cad4f8a35ea2c1ea16ff1ba98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#120;&#43;&#98;&#121;&#61;&#99;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"94\" style=\"vertical-align: -4px;\" \/> is a line. Each point on the line, an ordered pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c92a92f915d1f75c2c7a9f50c608cedd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/> is a solution to the equation. For a system of two equations with two variables, we graph two lines. Then we can see that all the points that are solutions to each equation form a line. And, by finding what the lines have in common, we\u2019ll find the solution to the system.<\/p>\n<p id=\"fs-id1167829690735\">Most linear equations in one variable have one solution, but we saw that some equations, called contradictions, have no solutions and for other equations, called identities, all numbers are solutions<\/p>\n<p id=\"fs-id1167833311012\">We know when we solve a system of two linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167829598038\" data-alt=\"Figure shows three graphs. In the first one, two lines intersect. Intersecting lines have one point in common. There is one solution to this system. The graph is labeled Consistent Independent. In the second graph, two lines are parallel. Parallel lines have no points in common. There is no solution to this system. The graph is labeled inconsistent. In the third graph, there is just one line. Both equations give the same line. Because we have just one line, there are infinitely many solutions. It is labeled consistent dependent.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows three graphs. In the first one, two lines intersect. Intersecting lines have one point in common. There is one solution to this system. The graph is labeled Consistent Independent. In the second graph, two lines are parallel. Parallel lines have no points in common. There is no solution to this system. The graph is labeled inconsistent. In the third graph, there is just one line. Both equations give the same line. Because we have just one line, there are infinitely many solutions. It is labeled consistent dependent.\" \/><\/span><\/p>\n<p id=\"fs-id1167836608495\">Similarly, for a linear equation with three variables <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c9ccc8755030d27b0619ae3c85176458_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#120;&#43;&#98;&#121;&#43;&#99;&#122;&#61;&#100;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"133\" style=\"vertical-align: -4px;\" \/> every solution to the equation is an ordered triple, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d846d38c62afb12b0640750a85b1cb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#44;&#122;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/> that makes the equation true.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836321732\">\n<div data-type=\"title\">Linear Equation in Three Variables<\/div>\n<p id=\"fs-id1167833129160\">A linear equation with three variables, where <em data-effect=\"italics\">a, b, c,<\/em> and <em data-effect=\"italics\">d<\/em> are real numbers and <em data-effect=\"italics\">a, b<\/em>, and <em data-effect=\"italics\">c<\/em> are not all 0, is of the form<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836601622\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ba0cc5a46c3dfb19993706fbbd742fd1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#120;&#43;&#98;&#121;&#43;&#99;&#122;&#61;&#100;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167833047411\">Every solution to the equation is an ordered triple, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d846d38c62afb12b0640750a85b1cb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#44;&#122;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/> that makes the equation true.<\/p>\n<\/div>\n<p id=\"fs-id1167836502713\">All the points that are solutions to one equation form a plane in three-dimensional space. And, by finding what the planes have in common, we\u2019ll find the solution to the system.<\/p>\n<p id=\"fs-id1167829809276\">When we solve a system of three linear equations represented by a graph of three planes in space, there are three possible cases.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167829690706\" data-alt=\"Eight figures are shown. The first one shows three intersecting planes with one point in common. It is labeled Consistent system and Independent equations. The second figure has three parallel planes with no points in common. It is labeled Inconsistent system. In the third figure two planes are coincident and parallel to the third plane. The planes have no points in common. In the fourth figure, two planes are parallel and each intersects the third plane. The planes have no points in common. In the fifth figure, each plane intersects the other two, but all three share no points. The planes have no points in common. In the sixth figure, three planes intersect in one line. There is just one line, so there are infinitely many solutions. In the seventh figure, two planes are coincident and intersect the third plane in a line. There is just one line, so there are infinitely many solutions. In the last figure, three planes are coincident. There is just one plane, so there are infinitely many solutions.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_002h_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Eight figures are shown. The first one shows three intersecting planes with one point in common. It is labeled Consistent system and Independent equations. The second figure has three parallel planes with no points in common. It is labeled Inconsistent system. In the third figure two planes are coincident and parallel to the third plane. The planes have no points in common. In the fourth figure, two planes are parallel and each intersects the third plane. The planes have no points in common. In the fifth figure, each plane intersects the other two, but all three share no points. The planes have no points in common. In the sixth figure, three planes intersect in one line. There is just one line, so there are infinitely many solutions. In the seventh figure, two planes are coincident and intersect the third plane in a line. There is just one line, so there are infinitely many solutions. In the last figure, three planes are coincident. There is just one plane, so there are infinitely many solutions.\" \/><\/span><span data-type=\"media\" id=\"fs-id1171790367628\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_002a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><span data-type=\"media\" id=\"fs-id1167826129302\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_002b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><span data-type=\"media\" id=\"fs-id1167829830109\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_002c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><span data-type=\"media\" id=\"fs-id1171792823031\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_002d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><span data-type=\"media\" id=\"fs-id1171790368377\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_002e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><span data-type=\"media\" id=\"fs-id1171790448102\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_002f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><span data-type=\"media\" id=\"fs-id1171792480661\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_002g_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<p id=\"fs-id1167836561085\">To solve a system of three linear equations, we want to find the values of the variables that are solutions to all three equations. In other words, we are looking for the ordered triple <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d846d38c62afb12b0640750a85b1cb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#44;&#122;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/> that makes all three equations true. These are called the <span data-type=\"term\">solutions of the system of three linear equations with three variables<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167829752800\">\n<div data-type=\"title\">Solutions of a System of Linear Equations with Three Variables<\/div>\n<p>Solutions of a system of equations are the values of the variables that make all the equations true. A solution is represented by an <span data-type=\"term\">ordered triple<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb4b614af92a847b38804f3b3de70610_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#44;&#122;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<p id=\"fs-id1167836440360\">To determine if an ordered triple is a solution to a system of three equations, we substitute the values of the variables into each equation. If the ordered triple makes all three equations true, it is a solution to the system.<\/p>\n<div data-type=\"example\" id=\"fs-id1167832929828\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829619748\">\n<div data-type=\"problem\" id=\"fs-id1167836553422\">\n<p id=\"fs-id1167836333599\">Determine whether the ordered triple is a solution to the system: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5fdb188d8e721008d31cd0e42d89b60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#121;&#43;&#122;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#121;&#45;&#122;&#61;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#50;&#121;&#43;&#122;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"169\" style=\"vertical-align: -28px;\" \/><\/p>\n<p id=\"fs-id1167836494346\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c57ab48a8c5f37d3654db6cfa929a49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#49;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a68f4708199615e22b6d63cedd2f8ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#45;&#51;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829785610\">\n<p id=\"fs-id1167836392160\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836512836\" data-alt=\"The equations are x minus y plus z equals 2, 2x minus y minus z equals minus 6 and 2x plus 2y plus z equals minus 3. Substituting minus 2 for x, minus 1 for y and 3 for z into all three equations, we find that all three hold true. Hence, minus 2, minus 1, 3 is a solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_003_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are x minus y plus z equals 2, 2x minus y minus z equals minus 6 and 2x plus 2y plus z equals minus 3. Substituting minus 2 for x, minus 1 for y and 3 for z into all three equations, we find that all three hold true. Hence, minus 2, minus 1, 3 is a solution.\" \/><\/span><\/p>\n<p id=\"fs-id1167833382620\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167825009206\" data-alt=\"The equations are x minus y plus z equals 2, 2x minus y minus z equals minus 6 and 2x plus 2y plus z equals minus 3. Substituting minus minus 4 for x, minus 3 for y and 4 for z into all three equations, we find that all three hold true. Hence, minus 4, minus 3, 4 is not a solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are x minus y plus z equals 2, 2x minus y minus z equals minus 6 and 2x plus 2y plus z equals minus 3. Substituting minus minus 4 for x, minus 3 for y and 4 for z into all three equations, we find that all three hold true. Hence, minus 4, minus 3, 4 is not a solution.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836572810\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836528199\">\n<p id=\"fs-id1167829744537\">Determine whether the ordered triple is a solution to the system: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9f93cf5e14fa061ecce8d922ee6cb9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#43;&#121;&#43;&#122;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#50;&#121;&#43;&#122;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#121;&#43;&#50;&#122;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"160\" style=\"vertical-align: -28px;\" \/><\/p>\n<p id=\"fs-id1167829621248\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7140185a3cb7fd08f500f5c5fb0c90d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#51;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-561a857284077ded5f1b5e98f66f9188_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#49;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836622967\">\n<p id=\"fs-id1167822971291\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836628976\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833139268\">\n<div data-type=\"problem\" id=\"fs-id1167829599455\">\n<p id=\"fs-id1167829690831\">Determine whether the ordered triple is a solution to the system: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb0d039b121cc72af2b9f17fc4244449_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#51;&#121;&#43;&#122;&#61;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#51;&#120;&#45;&#121;&#45;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#50;&#121;&#43;&#51;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"164\" style=\"vertical-align: -28px;\" \/><\/p>\n<p id=\"fs-id1167836650229\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf11aeb7a6ea31875aebc7a128e203ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-995970456479c9b8a3721f38892f8a70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829714944\">\n<p id=\"fs-id1167836546570\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836293725\">\n<h3 data-type=\"title\">Solve a System of Linear Equations with Three Variables<\/h3>\n<p id=\"fs-id1167836287881\">To solve a system of linear equations with three variables, we basically use the same techniques we used with systems that had two variables. We start with two pairs of equations and in each pair we eliminate the same variable. This will then give us a system of equations with only two variables and then we know how to solve that system!<\/p>\n<p id=\"fs-id1167833023923\">Next, we use the values of the two variables we just found to go back to the original equation and find the third variable. We write our answer as an ordered triple and then check our results.<\/p>\n<div data-type=\"example\" id=\"fs-id1167833350276\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Solve a System of Equations With Three Variables by Elimination<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836729945\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167829712854\">Solve the system by elimination: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c0b11cd24065530ebfcf0b5b658e2d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#50;&#121;&#43;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#121;&#43;&#122;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#52;&#121;&#43;&#51;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"178\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833369162\"><span data-type=\"media\" id=\"fs-id1167833335338\" data-alt=\"The equations are x minus 2y plus z equals 3, 2x plus y plus z equals 4 and 3x plus 4y plus 3z equals minus 1. Step 1 is to write the equations in standard form. They are. If any coefficients are fractions, clear them. There are none.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_005a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are x minus 2y plus z equals 3, 2x plus y plus z equals 4 and 3x plus 4y plus 3z equals minus 1. Step 1 is to write the equations in standard form. They are. If any coefficients are fractions, clear them. There are none.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167825830021\" data-alt=\"Step 2 is to eliminate the same variable from two equations. Decide which variable you will eliminate. We can eliminate the y\u2019s from equations 1 and 2 by multiplying equation 2 by 2. Work with a pair of equations to eliminate the chosen variable. Multiply one or both equations so that the coefficients of that variable are opposites. Add the equations resulting from Step 2 to eliminate one variable. The new equation we get is 5x plus 3z equals 11.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_005b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to eliminate the same variable from two equations. Decide which variable you will eliminate. We can eliminate the y\u2019s from equations 1 and 2 by multiplying equation 2 by 2. Work with a pair of equations to eliminate the chosen variable. Multiply one or both equations so that the coefficients of that variable are opposites. Add the equations resulting from Step 2 to eliminate one variable. The new equation we get is 5x plus 3z equals 11.\" \/><\/span><span data-type=\"media\" data-alt=\"Step 3 is to repeat step 2 using two other equations and eliminate the same variable as in step 2. We can again eliminate the y\u2019s using the equations 1, 3 by multiplying equation 1 by 2. Add the new equations and the result will be 5x plus 5z equals 5.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_005c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to repeat step 2 using two other equations and eliminate the same variable as in step 2. We can again eliminate the y\u2019s using the equations 1, 3 by multiplying equation 1 by 2. Add the new equations and the result will be 5x plus 5z equals 5.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836292792\" data-alt=\"Step 4. The two new equations form a system of two equations with two variables. Solve this system. Eliminating x, we get z equal to minus 3. Substituting this in one of the new equations, we get x equal to 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_005d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4. The two new equations form a system of two equations with two variables. Solve this system. Eliminating x, we get z equal to minus 3. Substituting this in one of the new equations, we get x equal to 4.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167826172504\" data-alt=\"Step 5 is to use the values of the two variables found in step 4 to find the third variable. Substituting values of x and z in one of the original equations, we get y equal to minus 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_005e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to use the values of the two variables found in step 4 to find the third variable. Substituting values of x and z in one of the original equations, we get y equal to minus 1.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836611284\" data-alt=\"Step 6 is to write the solution as an ordered triple 4, minus 1, minus 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_005f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 6 is to write the solution as an ordered triple 4, minus 1, minus 3.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167829790585\" data-alt=\"Step 7 is to check that the ordered triple is a solution to all three original equations. It makes all three equations true.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_005g_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 7 is to check that the ordered triple is a solution to all three original equations. It makes all three equations true.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829752373\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836621335\">\n<div data-type=\"problem\" id=\"fs-id1167829908545\">\n<p id=\"fs-id1167830077067\">Solve the system by elimination: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-21af0efb18b1178eee37eef30d7831c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#43;&#121;&#45;&#122;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#51;&#121;&#45;&#50;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#45;&#121;&#45;&#51;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"164\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833021838\">\n<p id=\"fs-id1167822193559\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4085f8769af10521ea3c6832647d40f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#49;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836295067\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836356451\">\n<div data-type=\"problem\" id=\"fs-id1167833076732\">\n<p id=\"fs-id1167822890955\">Solve the system by elimination: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39afd03ecd497e948033361319ea534f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#120;&#43;&#121;&#43;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#120;&#45;&#50;&#121;&#43;&#122;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#45;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"169\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829744332\">\n<p id=\"fs-id1167826170389\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-85b46fbd067fff4ddd51345abc8b51c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#51;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836389649\">The steps are summarized here.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836665602\" class=\"howto\">\n<div data-type=\"title\">Solve a system of linear equations with three variables.<\/div>\n<ol id=\"fs-id1167836598867\" type=\"1\" class=\"stepwise\">\n<li>Write the equations in standard form\n<ul id=\"fs-id1167836728212\" data-bullet-style=\"bullet\">\n<li>If any coefficients are fractions, clear them.<\/li>\n<\/ul>\n<\/li>\n<li>Eliminate the same variable from two equations.\n<ul id=\"fs-id1167836706262\" data-bullet-style=\"bullet\">\n<li>Decide which variable you will eliminate.<\/li>\n<li>Work with a pair of equations to eliminate the chosen variable.<\/li>\n<li>Multiply one or both equations so that the coefficients of that variable are opposites.<\/li>\n<li>Add the equations resulting from Step 2 to eliminate one variable<\/li>\n<\/ul>\n<\/li>\n<li>Repeat Step 2 using two other equations and eliminate the same variable as in Step 2.<\/li>\n<li>The two new equations form a system of two equations with two variables. Solve this system.<\/li>\n<li>Use the values of the two variables found in Step 4 to find the third variable.<\/li>\n<li>Write the solution as an ordered triple.<\/li>\n<li>Check that the ordered triple is a solution to <strong data-effect=\"bold\">all three<\/strong> original equations.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167833050653\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167832936787\">\n<div data-type=\"problem\" id=\"fs-id1167833020621\">\n<p id=\"fs-id1167833381373\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b3ec88b167f86dd88e1ea9cea4f44d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#52;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#121;&#43;&#50;&#122;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#61;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"138\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829593667\">\n<div data-type=\"equation\" id=\"fs-id1171784163338\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65437cfec61d66677d923fe23c09c122_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#52;&#122;&#61;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#121;&#43;&#50;&#122;&#61;&#45;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#61;&#45;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"156\" style=\"vertical-align: -28px;\" \/><\/div>\n<p id=\"fs-id1167836611848\">We can eliminate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> from equations (1) and (2) by multiplying equation (2) by 2 and then adding the resulting equations.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167826206324\" data-alt=\"The equations are 3 x minus 4 equals 0, 3y plus 2 z equals minus 3 and 2 x plus 3 y equals minus 5. Multiply equation 2 by 2 and add to equation 1. We get 3 x plus 6 y equals minus 6.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_006a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are 3 x minus 4 equals 0, 3y plus 2 z equals minus 3 and 2 x plus 3 y equals minus 5. Multiply equation 2 by 2 and add to equation 1. We get 3 x plus 6 y equals minus 6.\" \/><\/span><\/p>\n<p id=\"fs-id1167829784503\">Notice that equations (3) and (4) both have the variables <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>. We will solve this new system for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167829906127\" data-alt=\"Multiply equation 3 by minus 2 and add that to equation 4. We get x equal to minus 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_006b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Multiply equation 3 by minus 2 and add that to equation 4. We get x equal to minus 4.\" \/><\/span><\/p>\n<p id=\"fs-id1167829810622\">To solve for <em data-effect=\"italics\">y<\/em>, we substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3dc975a98ccada6f136856736d7df06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -1px;\" \/> into equation (3).<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167833369395\" data-alt=\"Substitute minus 4 into equation 3 and solve for y. We get y equal to 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_006c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Substitute minus 4 into equation 3 and solve for y. We get y equal to 1.\" \/><\/span><\/p>\n<p id=\"fs-id1167836714302\">We now have <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3dc975a98ccada6f136856736d7df06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4acfdc8013815a0ea0c7059786c6402_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/> We need to solve for <em data-effect=\"italics\">z<\/em>. We can substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3dc975a98ccada6f136856736d7df06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -1px;\" \/> into equation (1) to find <em data-effect=\"italics\">z<\/em>.<\/p>\n<p><span data-type=\"media\" data-alt=\"Substituting minus 4 into equation 1 for x, we get z equal to minus 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_006d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Substituting minus 4 into equation 1 for x, we get z equal to minus 3.\" \/><\/span><\/p>\n<p id=\"fs-id1167832978282\">We write the solution as an ordered triple. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61c319eadef0b6ea3db36b4b2d4adfdb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#49;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167836573935\">We check that the solution makes all three equations true.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1de9ba1e860fddc32d4ae1076f35cbb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#120;&#45;&#52;&#122;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#121;&#43;&#50;&#122;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#51;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#120;&#43;&#51;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#53;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#115;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#32;&#105;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#49;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"101\" width=\"729\" style=\"vertical-align: -62px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832940114\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167833382031\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01104d17bfb7da153316fa677c08cd6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#52;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#121;&#43;&#51;&#122;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"138\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836704561\">\n<p id=\"fs-id1167836620807\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a0f64124d85580a7a8e65bd45826c16b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#52;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167825782333\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836627315\">\n<div data-type=\"problem\" id=\"fs-id1167833382121\">\n<p id=\"fs-id1167833138643\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-08fd797cef364ec311e126aa25720633_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#120;&#45;&#51;&#122;&#61;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#121;&#43;&#50;&#122;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#52;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"138\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836434358\">\n<p id=\"fs-id1167836522895\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb18122f870ac76299b541f155738801_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#51;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833138169\">When we solve a system and end up with no variables and a false statement, we know there are no solutions and that the system is inconsistent. The next example shows a system of equations that is inconsistent.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829704831\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836576093\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167829878516\">Solve the system of equations: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-71c10b6eeef3cdb9d205651a40a4d786_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#50;&#121;&#45;&#51;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#51;&#121;&#43;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#121;&#45;&#50;&#122;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"169\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836624752\">\n<div data-type=\"equation\" id=\"fs-id1171791597904\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c5fef21e487fc60347f31f680752cfb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#50;&#121;&#45;&#51;&#122;&#61;&#45;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#51;&#121;&#43;&#122;&#61;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#121;&#45;&#50;&#122;&#61;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"187\" style=\"vertical-align: -28px;\" \/><\/div>\n<p>Use equation (1) and (2) to eliminate <em data-effect=\"italics\">z<\/em>.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167833021780\" data-alt=\"The equations are x plus 2y minus 3z equals minus 1, x minus 3y plus z equals 1 and 2x minus y minus 2z equals 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_007a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are x plus 2y minus 3z equals minus 1, x minus 3y plus z equals 1 and 2x minus y minus 2z equals 2.\" \/><\/span><\/p>\n<p>Use (2) and (3) to eliminate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> again.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167829719422\" data-alt=\"Multiplying equation 2 by 3 and adding it to equation 1, we get equation 4, 4x minus 7y equals 2. Multiplying equation 2 by 2 and adding it to equation 3, we get equation 5, 4x minus 7y equals 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_007b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Multiplying equation 2 by 3 and adding it to equation 1, we get equation 4, 4x minus 7y equals 2. Multiplying equation 2 by 2 and adding it to equation 3, we get equation 5, 4x minus 7y equals 4.\" \/><\/span><\/p>\n<p id=\"fs-id1167836521885\">Use (4) and (5) to eliminate a variable.<\/p>\n<p><span data-type=\"media\" data-alt=\"Equations 4 and 5 both have 2 variables. Multiply equation 5 by minus 1 and add it to equation 4. We get 0 equal to minus 2, which is false.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_007c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Equations 4 and 5 both have 2 variables. Multiply equation 5 by minus 1 and add it to equation 4. We get 0 equal to minus 2, which is false.\" \/><\/span><\/p>\n<p id=\"fs-id1167824731658\">There is no solution.<\/p>\n<p>We are left with a false statement and this tells us the system is inconsistent and has no solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836407993\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836334898\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836755457\">Solve the system of equations: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b669f86ce97c922e3bd333ee2e576af7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#50;&#121;&#43;&#54;&#122;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#120;&#43;&#121;&#45;&#50;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#52;&#121;&#45;&#50;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"160\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836788456\">\n<p id=\"fs-id1167829984082\">no solution<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829791296\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167833057146\">\n<p>Solve the system of equations: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29c4a4df4d656bf109f97f7205b6ef8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#45;&#50;&#121;&#43;&#51;&#122;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#45;&#51;&#121;&#43;&#50;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#120;&#43;&#51;&#121;&#45;&#55;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"178\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836574853\">\n<p id=\"fs-id1167836456023\">no solution<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167830004605\">When we solve a system and end up with no variables but a true statement, we know there are infinitely many solutions. The system is consistent with dependent equations. Our solution will show how two of the variables depend on the third.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836510584\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836398862\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836534914\">Solve the system of equations: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-011d385582c4d047b32ff4db72341def_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#50;&#121;&#45;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#55;&#121;&#43;&#52;&#122;&#61;&#49;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#51;&#121;&#43;&#122;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"173\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836286713\">\n<div data-type=\"equation\" id=\"fs-id1171791735385\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-74cd4436bbd505e698e76588084828f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#50;&#121;&#45;&#122;&#61;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#55;&#121;&#43;&#52;&#122;&#61;&#49;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#51;&#121;&#43;&#122;&#61;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"191\" style=\"vertical-align: -28px;\" \/><\/div>\n<p id=\"fs-id1167833139042\">Use equation (1) and (3) to eliminate <em data-effect=\"italics\">x<\/em>.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836508275\" data-alt=\"The equations are x plus 2y minus z equals 1, 2x plus 7y plus 4z equals 11 and x plus 3y plus z equals 4. Multiply equation 1 with minus 1 and add it to equation 3. We get equation 4, y plus 2z equals 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_008a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are x plus 2y minus z equals 1, 2x plus 7y plus 4z equals 11 and x plus 3y plus z equals 4. Multiply equation 1 with minus 1 and add it to equation 3. We get equation 4, y plus 2z equals 3.\" \/><\/span><\/p>\n<p id=\"fs-id1167833022192\">Use equation (1) and (2) to eliminate <em data-effect=\"italics\">x<\/em> again.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167826171290\" data-alt=\"Multiply equation 1 with minus 2 and add it to equation 2. We get equation 5, 3y plus 6z equals 9.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_008b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Multiply equation 1 with minus 2 and add it to equation 2. We get equation 5, 3y plus 6z equals 9.\" \/><\/span><\/p>\n<p id=\"fs-id1167836327517\">Use equation (4) and (5) to eliminate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1171789696361\" data-alt=\"Multiply equation 4 with minus 3 and add it to equation 5. We get 0 equal to 0. There are infinite many solutions. Solving equation 4 for y, we get y equal to minus 2z plus 3. Substituting this into equation 1, we get x equal to 5z minus 5. The true statement 0 equal to 0 tells us that this is a dependent system that has infinitely many solutions. The solutions are of the form x, y, z where x is 5z minus 5, y is minus 2z plus 3 and z is any real number.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_008c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Multiply equation 4 with minus 3 and add it to equation 5. We get 0 equal to 0. There are infinite many solutions. Solving equation 4 for y, we get y equal to minus 2z plus 3. Substituting this into equation 1, we get x equal to 5z minus 5. The true statement 0 equal to 0 tells us that this is a dependent system that has infinitely many solutions. The solutions are of the form x, y, z where x is 5z minus 5, y is minus 2z plus 3 and z is any real number.\" \/><\/span><\/p>\n<table id=\"fs-id1167824774013\" class=\"unnumbered unstyled\" summary=\"\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">There are infinitely many solutions.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve equation (4) for <em data-effect=\"italics\">y<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Represent the solution showing how <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> are dependent on <em data-effect=\"italics\">z<\/em>.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31f93c2a80437276c91a4531f8c01fdb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#43;&#50;&#122;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#50;&#122;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"157\" style=\"vertical-align: -15px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use equation (1) to solve for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b3cc14a558178c52eb674ab7bf1d044_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#43;&#50;&#121;&#45;&#122;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"112\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2fcdebfb437b7b70460ccae4fd1b59cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#50;&#122;&#43;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4575e63fd7cad0a35d3f08d541e5ad6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#43;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#122;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#122;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#45;&#52;&#122;&#43;&#54;&#45;&#122;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#45;&#53;&#122;&#43;&#54;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#53;&#122;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"80\" width=\"244\" style=\"vertical-align: -33px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836360117\">The true statement <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4595e14933eddbeae2bbaa66463fcca6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> tells us that this is a dependent system that has infinitely many solutions. The solutions are of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d846d38c62afb12b0640750a85b1cb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#44;&#122;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-52648904bd16d945ae0610cd14607d69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#53;&#122;&#45;&#53;&#59;&#121;&#61;&#45;&#50;&#122;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"185\" style=\"vertical-align: -4px;\" \/>and <em data-effect=\"italics\">z<\/em> is any real number.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829907255\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829796752\">\n<div data-type=\"problem\" id=\"fs-id1167829718219\">\n<p id=\"fs-id1167836492090\">Solve the system by equations: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec74736d43d54250b272d87f9b76a440_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#45;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#52;&#121;&#45;&#50;&#122;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#54;&#121;&#45;&#51;&#122;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"164\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829861781\">\n<p id=\"fs-id1167836531064\">infinitely many solutions<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2e754cad2201cb652dd306481046dfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#51;&#44;&#122;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4dde71aebc0159680c099c1e3f6751f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#122;&#45;&#51;&#59;&#121;&#61;&#51;&#59;&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"140\" style=\"vertical-align: -4px;\" \/> is any real number<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836530163\">\n<div data-type=\"problem\" id=\"fs-id1167836339987\">\n<p id=\"fs-id1167829719789\">Solve the system by equations: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-302dbe159958886bf645404520030fb2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#121;&#45;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#120;&#43;&#50;&#121;&#45;&#51;&#122;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#50;&#121;&#45;&#55;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"183\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836666754\">\n<p id=\"fs-id1167833086865\">infinitely many solutions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d846d38c62afb12b0640750a85b1cb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#44;&#122;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/> where<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6587d829e514a859d2bb753d9e633c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#53;&#122;&#45;&#50;&#59;&#121;&#61;&#52;&#122;&#45;&#51;&#59;&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"188\" style=\"vertical-align: -4px;\" \/> is any real number<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836539512\">\n<h3 data-type=\"title\">Solve Applications using Systems of Linear Equations with Three Variables<\/h3>\n<p id=\"fs-id1167833086461\">Applications that are modeled by a systems of equations can be solved using the same techniques we used to solve the systems. Many of the application are just extensions to three variables of the types we have solved earlier.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836622870\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167833227340\">\n<div data-type=\"problem\" id=\"fs-id1167829624534\">\n<p>The community college theater department sold three kinds of tickets to its latest play production. The adult tickets sold for ?15, the student tickets for ?10 and the child tickets for ?8. The theater department was thrilled to have sold 250 tickets and brought in ?2,825 in one night. The number of student tickets sold is twice the number of adult tickets sold. How many of each type did the department sell?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829693376\">\n<table id=\"fs-id1167829812971\" class=\"unnumbered unstyled can-break\" summary=\"We will use a chart to organize information. The type is adult, student and child. The numbers for these are x, y and z respectively and the total is 250. The values for these are 15, 10 and 8 respectively. The total values are 15x, 10y and 8z respectively, the total being 2825. Number of students is twice number of adults. So y is 2x. We rewrite as 2x minus y equals 0. The system of equations is x plus y plus z equals 250, 15x plus 10y plus 8z is 2825 and minus 2x plus y is 0. Multiply equation 1 by minus 8 and add it to equation 2. We get 7x plus 2y equals 825. Multiply equation 3 with minus 2 and add it to equation 4. Solving for x, we get x equal to 75. Substituting this into equation 3, we get y equal to 150. Substituting values of x and y in equation 1, we get z equal to 25. The theater department sold 75 adult tickets, 150 student tickets, and 25 child tickets.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">We will use a chart to organize the information.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836693208\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Number of students is twice number of adults.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"bottom\" data-align=\"left\">Rewrite the equation in standard form.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e92ba9a1b2f1bc8505737523cdf6301_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#120;&#45;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"115\" style=\"vertical-align: -15px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836449342\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_009b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use equations (1) and (2) to eliminate <em data-effect=\"italics\">z<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_009c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use (3) and (4) to eliminate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62f853fa6f372493298c507883a9f490_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836545039\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_009d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb715937a1af7a6a41bcd777df4c2286_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#8195;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#55;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"115\" style=\"vertical-align: 0px;\" \/> adult tickets<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use equation (3) to find <em data-effect=\"italics\">y<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b7b5784ff06b905d4b138b74a95bd23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#50;&#120;&#43;&#121;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ab98110c9cb84311da6218cbd16fae05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#55;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4913463ca48533b6eda8279410182915_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#53;&#48;&#43;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#53;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#116;&#117;&#100;&#101;&#110;&#116;&#32;&#116;&#105;&#99;&#107;&#101;&#116;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"277\" style=\"vertical-align: -26px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use equation (1) to find <em data-effect=\"italics\">z<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b46cfa64e106861073e61932592a6ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#43;&#121;&#43;&#122;&#61;&#50;&#53;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute in the values<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27024574e6a11e3790d276c79f121557_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#55;&#53;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#61;&#49;&#53;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\">\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-801db9467ecf107b9f21a287e8f0235e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#55;&#53;&#43;&#49;&#53;&#48;&#43;&#122;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#53;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#50;&#53;&#43;&#122;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#53;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#122;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#104;&#105;&#108;&#100;&#32;&#116;&#105;&#99;&#107;&#101;&#116;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"257\" style=\"vertical-align: -22px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the solution.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The theater department sold 75 adult tickets,<\/p>\n<div data-type=\"newline\"><\/div>\n<p>150 student tickets, and 25 child tickets.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833279849\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829651324\">\n<div data-type=\"problem\" id=\"fs-id1167836449776\">\n<p id=\"fs-id1167832976579\">The community college fine arts department sold three kinds of tickets to its latest dance presentation. The adult tickets sold for ?20, the student tickets for ?12 and the child tickets for ?10.The fine arts department was thrilled to have sold 350 tickets and brought in ?4,650 in one night. The number of child tickets sold is the same as the number of adult tickets sold. How many of each type did the department sell?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829811278\">\n<p id=\"fs-id1167829715922\">The fine arts department sold 75 adult tickets, 200 student tickets, and 75 child tickets.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836330695\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167829904743\">\n<p>The community college soccer team sold three kinds of tickets to its latest game. The adult tickets sold for ?10, the student tickets for ?8 and the child tickets for ?5. The soccer team was thrilled to have sold 600 tickets and brought in ?4,900 for one game. The number of adult tickets is twice the number of child tickets. How many of each type did the soccer team sell?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836627190\">\n<p id=\"fs-id1167825872620\">The soccer team sold 200 adult tickets, 300 student tickets, and 100 child tickets.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836509791\" class=\"media-2\">\n<p id=\"fs-id1167829831129\">Access this online resource for additional instruction and practice with solving a linear system in three variables with no or infinite solutions.<\/p>\n<ul id=\"fs-id1169146669576\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37linsys3var\">Solving a Linear System in Three Variables with No or Infinite Solutions<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37variableapp\">3 Variable Application<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167836774761\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Linear Equation in Three Variables:<\/strong> A linear equation with three variables, where <em data-effect=\"italics\">a, b, c<\/em>, and <em data-effect=\"italics\">d<\/em> are real numbers and <em data-effect=\"italics\">a, b,<\/em> and <em data-effect=\"italics\">c<\/em> are not all 0, is of the form\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167829743566\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ba0cc5a46c3dfb19993706fbbd742fd1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#120;&#43;&#98;&#121;&#43;&#99;&#122;&#61;&#100;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"newline\"><\/div>\n<p> Every solution to the equation is an ordered triple, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d846d38c62afb12b0640750a85b1cb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#44;&#122;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/> that makes the equation true.<\/li>\n<li><strong data-effect=\"bold\">How to solve a system of linear equations with three variables.<\/strong>\n<ol id=\"fs-id1167833365883\" type=\"1\" class=\"stepwise\">\n<li>Write the equations in standard form\n<div data-type=\"newline\"><\/div>\n<p> If any coefficients are fractions, clear them.<\/li>\n<li>Eliminate the same variable from two equations.\n<div data-type=\"newline\"><\/div>\n<p> Decide which variable you will eliminate.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> Work with a pair of equations to eliminate the chosen variable.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> Multiply one or both equations so that the coefficients of that variable are opposites.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> Add the equations resulting from Step 2 to eliminate one variable<\/li>\n<li>Repeat Step 2 using two other equations and eliminate the same variable as in Step 2.<\/li>\n<li>The two new equations form a system of two equations with two variables. Solve this system.<\/li>\n<li>Use the values of the two variables found in Step 4 to find the third variable.<\/li>\n<li>Write the solution as an ordered triple.<\/li>\n<li>Check that the ordered triple is a solution to <strong data-effect=\"bold\">all three<\/strong> original equations.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167829906194\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167829634077\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167830096283\"><strong data-effect=\"bold\">Determine Whether an Ordered Triple is a Solution of a System of Three Linear Equations with Three Variables<\/strong><\/p>\n<p id=\"fs-id1167829936718\">In the following exercises, determine whether the ordered triple is a solution to the system.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836501642\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833086359\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f04b99ed6e9c81e2f3a702a5d564797a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#45;&#54;&#121;&#43;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#52;&#121;&#45;&#51;&#122;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#45;&#50;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"152\" style=\"vertical-align: -28px;\" \/><\/p>\n<p id=\"fs-id1167836706047\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed349cfc50391f6bcebabc6c89669d23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#49;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b0593d8517bda6a6bc9c092c9720e0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#51;&#44;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836510397\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167824748848\">\n<p id=\"fs-id1167829651306\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2811370938f15b934bd50ecefa34c0fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#51;&#120;&#43;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#43;&#122;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#120;&#43;&#50;&#121;&#45;&#50;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#45;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"169\" style=\"vertical-align: -28px;\" \/><\/p>\n<p id=\"fs-id1167836287729\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36721313646ed01b2a664960041330d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#45;&#55;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7307d8b16707f014143c59fffe4de9a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#55;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836573198\">\n<p id=\"fs-id1167829807906\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> yes<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833136684\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167833021377\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd63c68ac599be51c7a9072f7aca2107_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#45;&#49;&#48;&#122;&#61;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#53;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"125\" style=\"vertical-align: -28px;\" \/><\/p>\n<p id=\"fs-id1167833046978\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d008ed721c0fe85350f50fdac61108fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#49;&#50;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"63\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9252f3a84f6e95d640166f78fbfd7f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836686650\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836686455\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a9d7fcc03a2750d621173c6fa2d355d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#51;&#121;&#45;&#122;&#61;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#51;&#121;&#43;&#122;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"147\" style=\"vertical-align: -28px;\" \/><\/p>\n<p id=\"fs-id1167829741854\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-797ad99adb3ef2fbd7a2283fd3745ab9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#44;&#53;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"70\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce7aa838e0429abb5822644b14b709b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"71\" style=\"vertical-align: -7px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167829696536\">\n<p id=\"fs-id1167829598072\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> yes<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829713554\"><strong data-effect=\"bold\">Solve a System of Linear Equations with Three Variables<\/strong><\/p>\n<p>In the following exercises, solve the system of equations.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167833327314\">\n<div data-type=\"problem\" id=\"fs-id1167836390323\">\n<p id=\"fs-id1167836738284\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7b8c22cbac528c0c4d951a87d9a1b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#53;&#120;&#43;&#50;&#121;&#43;&#122;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#51;&#120;&#45;&#121;&#43;&#50;&#122;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#45;&#51;&#122;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"157\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833086748\">\n<div data-type=\"problem\" id=\"fs-id1167824740545\">\n<p id=\"fs-id1167829872202\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65a9aa5c41925f40b586b954f991e6a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#54;&#120;&#45;&#53;&#121;&#43;&#50;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#121;&#45;&#52;&#122;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#51;&#121;&#43;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"156\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829749058\">\n<p id=\"fs-id1167836516626\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1835e86962c0e1a2cabe9a39f9410ce9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#53;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836392614\">\n<div data-type=\"problem\" id=\"fs-id1167824774411\">\n<p id=\"fs-id1167829594499\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-76598839a094c81107ecd02b8cb09442_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#45;&#53;&#121;&#43;&#51;&#122;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#121;&#43;&#52;&#122;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#51;&#121;&#43;&#50;&#122;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"157\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836493253\">\n<div data-type=\"problem\" id=\"fs-id1167836288151\">\n<p id=\"fs-id1167829717239\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c93dec45adb882a388cb73ef9708aaa9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#53;&#120;&#45;&#51;&#121;&#43;&#50;&#122;&#61;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#121;&#45;&#122;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#50;&#121;&#43;&#50;&#122;&#61;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"166\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824764689\">\n<p id=\"fs-id1167836703915\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2bb0432c0fe0829cb5b01d3f58a4cf50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#49;&#50;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"77\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836477516\">\n<div data-type=\"problem\" id=\"fs-id1167836508453\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ba9dc063ea452d762aa7f3850fb9f1c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#53;&#121;&#43;&#52;&#122;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#120;&#43;&#50;&#121;&#43;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#45;&#50;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"152\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829878819\">\n<div data-type=\"problem\" id=\"fs-id1167836547728\">\n<p id=\"fs-id1167836691963\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46af7d0154576c52b63bbcb285e6c8e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#120;&#45;&#51;&#121;&#43;&#122;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#53;&#121;&#45;&#52;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#50;&#121;&#45;&#50;&#122;&#61;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"166\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826170426\">\n<p id=\"fs-id1167829930733\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a8bf7af7c2b74294bafbdf216e77260d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#53;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830013969\">\n<div data-type=\"problem\" id=\"fs-id1167836481012\">\n<p id=\"fs-id1167833025488\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e45d7718f0949c346f1a89da8ee46496_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#43;&#56;&#121;&#43;&#50;&#122;&#61;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#53;&#121;&#45;&#51;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#50;&#121;&#45;&#50;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"165\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832999524\">\n<div data-type=\"problem\" id=\"fs-id1167836532597\">\n<p id=\"fs-id1167829716811\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6756c4649ebcdecf0ecb323345eb407_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#49;&#49;&#120;&#43;&#57;&#121;&#43;&#50;&#122;&#61;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#55;&#120;&#43;&#53;&#121;&#43;&#51;&#122;&#61;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#43;&#51;&#121;&#43;&#122;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"175\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832925579\">\n<p id=\"fs-id1167829786231\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-83047e0a7e798a1f70ee8b86a4967cc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#51;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836689151\">\n<div data-type=\"problem\" id=\"fs-id1167824774450\">\n<p id=\"fs-id1167836515372\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38a37e66ed5e3a85617373470166d97a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#121;&#45;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#121;&#43;&#122;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#50;&#121;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#122;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"66\" width=\"168\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836614091\">\n<div data-type=\"problem\" id=\"fs-id1167829711744\">\n<p id=\"fs-id1167836705310\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4918ba518a7995b8c9a88eb456d168f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#121;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#121;&#43;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#121;&#43;&#50;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"66\" width=\"169\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836300586\">\n<p id=\"fs-id1167836415047\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-69982fdedc7a315890921d1c2b4ba397_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#45;&#57;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836598004\">\n<div data-type=\"problem\" id=\"fs-id1167833272219\">\n<p id=\"fs-id1167830077526\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-68e0ee5302b0366d9860529ccd719189_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#121;&#45;&#50;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#43;&#121;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#121;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"66\" width=\"171\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836510914\">\n<div data-type=\"problem\" id=\"fs-id1167836732591\">\n<p id=\"fs-id1167829787685\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d8018b9efc10cd2d2f2b997a4ffb0ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#121;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#122;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#121;&#45;&#52;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#121;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#122;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"66\" width=\"156\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829745800\">\n<p id=\"fs-id1167836597056\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a8474fb75c9f36c75231d626bf6e7e3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#52;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836362489\">\n<div data-type=\"problem\" id=\"fs-id1167836552028\">\n<p id=\"fs-id1167836585257\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7fad776d83f482d29bfa359f801a709b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#50;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#121;&#43;&#51;&#122;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#53;&#121;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"124\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829748029\">\n<div data-type=\"problem\" id=\"fs-id1167829905146\">\n<p id=\"fs-id1167836508175\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd958c5a425944a914f86a59bb431a44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#53;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#121;&#45;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#43;&#51;&#122;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"126\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829580059\">\n<p id=\"fs-id1167836549784\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5cccb83f742cb3d351a80d5b7c423823_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829807670\">\n<div data-type=\"problem\" id=\"fs-id1167833138383\">\n<p id=\"fs-id1167829686513\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bac1fd9dbb34b3d6893bb17cc3705dcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#121;&#43;&#51;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#120;&#43;&#51;&#121;&#61;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#55;&#120;&#43;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"126\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829740446\">\n<div data-type=\"problem\" id=\"fs-id1167833329016\">\n<p id=\"fs-id1167829989579\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0cd407da293f78735b7b33e61c00624a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#122;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#121;&#43;&#50;&#122;&#61;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#43;&#51;&#121;&#61;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"126\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836542169\">\n<p id=\"fs-id1167836387902\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-59f33043d348b9479472c49b84c012e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#48;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836532846\">\n<div data-type=\"problem\" id=\"fs-id1167829744480\">\n<p id=\"fs-id1167836545984\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a8d6b86bdc62745a20b9257fa7debc11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#120;&#45;&#51;&#121;&#43;&#50;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#120;&#43;&#51;&#121;&#45;&#55;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#50;&#121;&#43;&#51;&#122;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"165\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829811216\">\n<div data-type=\"problem\" id=\"fs-id1167836447777\">\n<p id=\"fs-id1167836299522\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2b49e0c145fd4e0f6bed865ff2d94223_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#50;&#121;&#43;&#50;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#120;&#43;&#121;&#45;&#122;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#121;&#43;&#122;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"147\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836664947\">\n<p id=\"fs-id1167833397142\">no solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829850954\">\n<div data-type=\"problem\" id=\"fs-id1167829844151\">\n<p id=\"fs-id1167833009058\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aea6601b106cf1465bade3cf7be311ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#51;&#121;&#43;&#122;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#121;&#43;&#122;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#52;&#121;&#43;&#50;&#122;&#61;&#50;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"161\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836524317\">\n<div data-type=\"problem\" id=\"fs-id1167829893475\">\n<p id=\"fs-id1167825791285\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a7d59703332d5e82347435fff2569f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#52;&#121;&#43;&#122;&#61;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#45;&#121;&#43;&#51;&#122;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#55;&#121;&#43;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"148\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833240113\">\n<p id=\"fs-id1167836507837\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02c2dd474068a85042ccce6f47831466_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#48;&#51;&#125;&#123;&#49;&#54;&#125;&#59;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#53;&#125;&#123;&#49;&#54;&#125;&#59;&#122;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#51;&#49;&#125;&#123;&#49;&#54;&#125;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"207\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829930558\">\n<div data-type=\"problem\" id=\"fs-id1167836292764\">\n<p id=\"fs-id1167836635633\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-898ffe5322657a8eca104425bc431ae5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#50;&#121;&#43;&#122;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#121;&#45;&#50;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#120;&#45;&#51;&#121;&#43;&#122;&#61;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"171\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836318721\">\n<div data-type=\"problem\" id=\"fs-id1167836318723\">\n<p id=\"fs-id1167825836140\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ade25ff4586fe539e9f604f5d8183e8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#45;&#50;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#120;&#45;&#51;&#121;&#43;&#122;&#61;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#50;&#121;&#43;&#122;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"171\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829593597\">\n<p id=\"fs-id1167829593599\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d846d38c62afb12b0640750a85b1cb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#44;&#122;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da492a7a1d7970c5a442660b447d2a24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#53;&#122;&#43;&#50;&#59;&#121;&#61;&#45;&#51;&#122;&#43;&#49;&#59;&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"202\" style=\"vertical-align: -4px;\" \/> is any real number<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833135136\">\n<div data-type=\"problem\" id=\"fs-id1167832961255\">\n<p id=\"fs-id1167832961257\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-56b45da8385f4c3ef2880137f979ffa6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#45;&#51;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#45;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#120;&#43;&#50;&#121;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"147\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829598896\">\n<div data-type=\"problem\" id=\"fs-id1167836693792\">\n<p id=\"fs-id1167836693795\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-776512881501bec376fe9292b38334cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#50;&#121;&#43;&#51;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#121;&#45;&#51;&#122;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#52;&#121;&#43;&#53;&#122;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"152\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833290526\">\n<p id=\"fs-id1167833290529\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d846d38c62afb12b0640750a85b1cb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#44;&#122;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6587d829e514a859d2bb753d9e633c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#53;&#122;&#45;&#50;&#59;&#121;&#61;&#52;&#122;&#45;&#51;&#59;&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"188\" style=\"vertical-align: -4px;\" \/> is any real number<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836448592\"><strong data-effect=\"bold\">Solve Applications using Systems of Linear Equations with Three Variables<\/strong><\/p>\n<p id=\"fs-id1167836501809\">In the following exercises, solve the given problem.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836501812\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836501814\">\n<p id=\"fs-id1167833114798\">The sum of the measures of the angles of a triangle is 180. The sum of the measures of the second and third angles is twice the measure of the first angle. The third angle is twelve more than the second. Find the measures of the three angles.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836561270\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836561272\">\n<p id=\"fs-id1167829712734\">The sum of the measures of the angles of a triangle is 180. The sum of the measures of the second and third angles is three times the measure of the first angle. The third angle is fifteen more than the second. Find the measures of the three angles.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836625011\">\n<p id=\"fs-id1167836625013\">42, 50, 58<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836556299\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836556301\">\n<p id=\"fs-id1167836556303\">After watching a major musical production at the theater, the patrons can purchase souvenirs. If a family purchases 4 t-shirts, the video, and 1 stuffed animal, their total is ?135.<\/p>\n<p id=\"fs-id1167836618962\">A couple buys 2 t-shirts, the video, and 3 stuffed animals for their nieces and spends ?115. Another couple buys 2 t-shirts, the video, and 1 stuffed animal and their total is ?85. What is the cost of each item?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836525390\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830077435\">\n<p id=\"fs-id1167830077437\">The church youth group is selling snacks to raise money to attend their convention. Amy sold 2 pounds of candy, 3 boxes of cookies and 1 can of popcorn for a total sales of ?65. Brian sold 4 pounds of candy, 6 boxes of cookies and 3 cans of popcorn for a total sales of ?140. Paulina sold 8 pounds of candy, 8 boxes of cookies and 5 cans of popcorn for a total sales of ?250. What is the cost of each item?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830077439\">\n<p id=\"fs-id1167836694284\">?20, ?5, ?10<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167829905105\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167829807546\">\n<div data-type=\"problem\" id=\"fs-id1167829807548\">\n<p>In your own words explain the steps to solve a system of linear equations with three variables by elimination.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833047500\">\n<div data-type=\"problem\" id=\"fs-id1167836518239\">\n<p id=\"fs-id1167836518241\">How can you tell when a system of three linear equations with three variables has no solution? Infinitely many solutions?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836578708\">\n<p id=\"fs-id1167836578710\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836387211\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167836477611\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836705926\" data-alt=\"This table has 4 columns, 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don\u2019t get it. The first row contains the following statements: determine whether an ordered triple is a solution of a system of three linear equations with three variables, solve a system of linear equations with three variables, solve applications using systems of linear equations with three variables. The remaining columns are blank.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_04_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns, 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don\u2019t get it. The first row contains the following statements: determine whether an ordered triple is a solution of a system of three linear equations with three variables, solve a system of linear equations with three variables, solve applications using systems of linear equations with three variables. The remaining columns are blank.\" \/><\/span><\/p>\n<p id=\"fs-id1167829596900\"><span class=\"token\">\u24d1<\/span> On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1167829719698\">\n<dt>solutions of a system of linear equations with three variables<\/dt>\n<dd id=\"fs-id1167825830106\">The solutions of a system of equations are the values of the variables that make all the equations true; a solution is represented by an ordered triple <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb4b614af92a847b38804f3b3de70610_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#44;&#122;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" style=\"vertical-align: -4px;\" \/><\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2513","chapter","type-chapter","status-publish","hentry"],"part":2305,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2513","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":0,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2513\/revisions"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/2305"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2513\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=2513"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=2513"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=2513"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=2513"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}