{"id":2574,"date":"2018-12-11T13:41:49","date_gmt":"2018-12-11T18:41:49","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-systems-of-equations-using-matrices\/"},"modified":"2018-12-11T13:41:49","modified_gmt":"2018-12-11T18:41:49","slug":"solve-systems-of-equations-using-matrices","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-systems-of-equations-using-matrices\/","title":{"raw":"Solve Systems of Equations Using Matrices","rendered":"Solve Systems of Equations Using Matrices"},"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>Write the augmented matrix for a system of equations<\/li><li>Use row operations on a matrix<\/li><li>Solve systems of equations using matrices<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167836546274\" class=\"be-prepared\"><p id=\"fs-id1167836578744\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167836704694\" type=\"1\"><li>Solve: \\(3\\left(x+2\\right)+4=4\\left(2x-1\\right)+9.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167836432956\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Solve: \\(0.25p+0.25\\left(x+4\\right)=5.20.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167836399284\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Evaluate when \\(x=-2\\) and \\(y=3\\text{:}\\phantom{\\rule{0.2em}{0ex}}2{x}^{2}-xy+3{y}^{2}.\\)<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><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836398812\"><h3 data-type=\"title\">Write the Augmented Matrix for a System of Equations<\/h3><p id=\"fs-id1167836731941\">Solving a system of equations can be a tedious operation where a simple mistake can wreak havoc on finding the solution. An alternative method which uses the basic procedures of elimination but with notation that is simpler is available. The method involves using a <span data-type=\"term\">matrix<\/span>. A matrix is a rectangular array of numbers arranged in rows and columns.<\/p><div data-type=\"note\" id=\"fs-id1167826205478\"><div data-type=\"title\">Matrix<\/div><p id=\"fs-id1167836508293\">A <strong data-effect=\"bold\">matrix<\/strong> is a rectangular array of numbers arranged in rows and columns.<\/p><p id=\"fs-id1167833048299\">A matrix with <em data-effect=\"italics\">m<\/em> rows and <em data-effect=\"italics\">n<\/em> columns has order \\(m\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}n.\\) The matrix on the left below has 2 rows and 3 columns and so it has order \\(2\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3.\\) We say it is a 2 by 3 matrix.<\/p><span data-type=\"media\" data-alt=\"Figure shows two matrices. The one on the left has the numbers minus 3, minus 2 and 2 in the first row and the numbers minus 1, 4 and 5 in the second row. The rows and columns are enclosed within brackets. Thus, it has 2 rows and 3 columns. It is labeled 2 cross 3 or 2 by 3 matrix. The matrix on the right is similar but with 3 rows and 4 columns. It is labeled 3 by 4 matrix.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows two matrices. The one on the left has the numbers minus 3, minus 2 and 2 in the first row and the numbers minus 1, 4 and 5 in the second row. The rows and columns are enclosed within brackets. Thus, it has 2 rows and 3 columns. It is labeled 2 cross 3 or 2 by 3 matrix. The matrix on the right is similar but with 3 rows and 4 columns. It is labeled 3 by 4 matrix.\"><\/span><p id=\"fs-id1167836693648\">Each number in the matrix is called an element or entry in the matrix.<\/p><\/div><p id=\"fs-id1167836508919\">We will use a matrix to represent a system of linear equations. We write each equation in standard form and the coefficients of the variables and the constant of each equation becomes a row in the matrix. Each column then would be the coefficients of one of the variables in the system or the constants. A vertical line replaces the equal signs. We call the resulting matrix the augmented matrix for the system of equations.<\/p><span data-type=\"media\" data-alt=\"The equations are 3x plus y equals minus 3 and 2x plus 3y equals 6. A 2 by 3 matrix is shown. The first row is 3, 1, minus 3. The second row is 2, 3, 6. The first column is labeled coefficients of x. The second column is labeled coefficients of y and the third is labeled constants.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_002_img_Errata.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are 3x plus y equals minus 3 and 2x plus 3y equals 6. A 2 by 3 matrix is shown. The first row is 3, 1, minus 3. The second row is 2, 3, 6. The first column is labeled coefficients of x. The second column is labeled coefficients of y and the third is labeled constants.\"><\/span><p id=\"fs-id1167829807716\">Notice the first column is made up of all the coefficients of <em data-effect=\"italics\">x<\/em>, the second column is the all the coefficients of <em data-effect=\"italics\">y<\/em>, and the third column is all the constants.<\/p><div data-type=\"example\" id=\"fs-id1167836598326\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829685776\"><div data-type=\"problem\" id=\"fs-id1167836510894\"><p id=\"fs-id1167836573732\">Write each system of linear equations as an augmented matrix:<\/p><p><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}5x-3y=-1\\hfill \\\\ y=2x-2\\hfill \\end{array}\\)<span class=\"token\">\u24d1<\/span>\\(\\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-id1167836610644\"><p id=\"fs-id1167836486232\"><span class=\"token\">\u24d0<\/span> The second equation is not in standard form. We rewrite the second equation in standard form.<\/p><div data-type=\"equation\" id=\"fs-id1167833056775\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{ccc}\\hfill y&amp; =\\hfill &amp; 2x-2\\hfill \\\\ \\hfill -2x+y&amp; =\\hfill &amp; -2\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167829853949\">We replace the second equation with its standard form. In the augmented matrix, the first equation gives us the first row and the second equation gives us the second row. The vertical line replaces the equal signs.<\/p><span data-type=\"media\" id=\"fs-id1167836664093\" data-alt=\"The equations are 3x plus y equals minus 3 and 2x plus 3y equals 6. A 2 by 3 matrix is shown. The first row is 3, 1, minus 3. The second row is 2, 3, 6. The first column is labeled coefficients of x. The second column is labeled coefficients of y and the third is labeled constants.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_003_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are 3x plus y equals minus 3 and 2x plus 3y equals 6. A 2 by 3 matrix is shown. The first row is 3, 1, minus 3. The second row is 2, 3, 6. The first column is labeled coefficients of x. The second column is labeled coefficients of y and the third is labeled constants.\"><\/span><p id=\"fs-id1167825836573\"><span class=\"token\">\u24d1<\/span> All three equations are in standard form. In the augmented matrix the first equation gives us the first row, the second equation gives us the second row, and the third equation gives us the third row. The vertical line replaces the equal signs.<\/p><span data-type=\"media\" id=\"fs-id1167822916222\" data-alt=\"The equations are 6x minus 5y plus 2z equals 3, 2x plus y minus 4z equals 5 and 3x minus 3y plus z equals minus 1. A 4 by 3 matrix is shown whose first row is 6, minus 5, 2, 3. Its second row is 2, 1, minus 4, 5. Its third row is 3, minus 3, 1 and minus 1. Its first three columns are labeled x, y and z respectively.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are 6x minus 5y plus 2z equals 3, 2x plus y minus 4z equals 5 and 3x minus 3y plus z equals minus 1. A 4 by 3 matrix is shown whose first row is 6, minus 5, 2, 3. Its second row is 2, 1, minus 4, 5. Its third row is 3, minus 3, 1 and minus 1. Its first three columns are labeled x, y and z respectively.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833059065\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829692719\"><div data-type=\"problem\" id=\"fs-id1167836774865\"><p id=\"fs-id1167829743626\">Write each system of linear equations as an augmented matrix:<\/p><p id=\"fs-id1167836522302\"><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}3x+8y=-3\\hfill \\\\ 2x=-5y-3\\hfill \\end{array}\\)<span class=\"token\">\u24d1<\/span>\\(\\left\\{\\begin{array}{c}2x-5y+3z=8\\hfill \\\\ 3x-y+4z=7\\hfill \\\\ x+3y+2z=-3\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836292577\"><p id=\"fs-id1167833310842\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left[\\begin{array}{ccccccc}3\\hfill &amp; &amp; &amp; 8\\hfill &amp; &amp; &amp; \\hfill -3\\\\ 2\\hfill &amp; &amp; &amp; 5\\hfill &amp; &amp; &amp; \\hfill -3\\end{array}\\right]\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left[\\begin{array}{cccccccccc}2\\hfill &amp; &amp; &amp; \\hfill -5&amp; &amp; &amp; 3\\hfill &amp; &amp; &amp; \\hfill 8\\\\ 3\\hfill &amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; 4\\hfill &amp; &amp; &amp; \\hfill 7\\\\ 1\\hfill &amp; &amp; &amp; \\hfill 3&amp; &amp; &amp; 2\\hfill &amp; &amp; &amp; \\hfill -3\\end{array}\\right]\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829829156\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833366592\"><div data-type=\"problem\"><p id=\"fs-id1167836546221\">Write each system of linear equations as an augmented matrix:<\/p><p id=\"fs-id1167829753078\"><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}11x=-9y-5\\hfill \\\\ 7x+5y=-1\\hfill \\end{array}\\)<span class=\"token\">\u24d1<\/span>\\(\\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-id1167836527130\"><p id=\"fs-id1167829598310\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left[\\begin{array}{ccccccc}11\\hfill &amp; &amp; &amp; 9\\hfill &amp; &amp; &amp; \\hfill -5\\\\ 7\\hfill &amp; &amp; &amp; 5\\hfill &amp; &amp; &amp; \\hfill -1\\end{array}\\right]\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left[\\begin{array}{cccccccccc}5\\hfill &amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill -5\\\\ 2\\hfill &amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill 4\\\\ 3\\hfill &amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill -7\\end{array}\\right]\\)<\/div><\/div><\/div><p id=\"fs-id1167833335409\">It is important as we solve systems of equations using matrices to be able to go back and forth between the system and the matrix. The next example asks us to take the information in the matrix and write the system of equations.<\/p><div data-type=\"example\" id=\"fs-id1167829908050\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829721061\"><div data-type=\"problem\" id=\"fs-id1167829579076\"><p id=\"fs-id1167829877945\">Write the system of equations that corresponds to the augmented matrix:<\/p><p>\\(\\left[\\begin{array}{ccccccccc}\\hfill 4&amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill 3&amp; &amp; \\\\ \\hfill 1&amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill -1&amp; &amp; \\\\ \\hfill -2&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill 3&amp; &amp; \\end{array}|\\begin{array}{ccc}&amp; &amp; \\hfill -1\\\\ &amp; &amp; \\hfill 2\\\\ &amp; &amp; \\hfill -4\\end{array}\\right].\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833335007\"><p>We remember that each row corresponds to an equation and that each entry is a coefficient of a variable or the constant. The vertical line replaces the equal sign. Since this matrix is a \\(4\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3\\), we know it will translate into a system of three equations with three variables.<\/p><span data-type=\"media\" id=\"fs-id1167823183143\" data-alt=\"A 3 by 4 matrix is shown. Its first row is 4, minus 3, 3, minus 1. Its second row is 1, 2, minus 1, 2. Its third row is minus 2, minus 1, 3, minus 4. The three equations are 4x minus 3y plus 3z equals minus 1, x plus 2y minus z equals 2 and minus 2x minus y plus 3z equals minus 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A 3 by 4 matrix is shown. Its first row is 4, minus 3, 3, minus 1. Its second row is 1, 2, minus 1, 2. Its third row is minus 2, minus 1, 3, minus 4. The three equations are 4x minus 3y plus 3z equals minus 1, x plus 2y minus z equals 2 and minus 2x minus y plus 3z equals minus 4.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167824755056\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167826188018\"><div data-type=\"problem\"><p id=\"fs-id1167836743351\">Write the system of equations that corresponds to the augmented matrix: \\(\\left[\\begin{array}{cccccccccc}1\\hfill &amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill 3\\\\ 2\\hfill &amp; &amp; &amp; \\hfill 1&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill 1\\\\ 4\\hfill &amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill 0\\end{array}\\right].\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167833350368\">\\(\\left\\{\\begin{array}{c}\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}x-y+2z=3\\hfill \\\\ 2x+y-2z=1\\hfill \\\\ \\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}4x-y+2z=0\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167824617356\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836497729\"><div data-type=\"problem\" id=\"fs-id1167836693299\"><p id=\"fs-id1167836691914\">Write the system of equations that corresponds to the augmented matrix: \\(\\left[\\begin{array}{cccccccccc}1\\hfill &amp; &amp; &amp; 1\\hfill &amp; &amp; &amp; \\hfill 1&amp; &amp; &amp; \\hfill 4\\\\ 2\\hfill &amp; &amp; &amp; 3\\hfill &amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill 8\\\\ 1\\hfill &amp; &amp; &amp; 1\\hfill &amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill 3\\end{array}\\right].\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829717165\"><p id=\"fs-id1167836293044\">\\(\\left\\{\\begin{array}{c}\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}x+y+z=4\\hfill \\\\ 2x+3y-z=8\\hfill \\\\ \\phantom{\\rule{0.8em}{0ex}}x+y-z=3\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\"><h3 data-type=\"title\">Use Row Operations on a Matrix<\/h3><p>Once a system of equations is in its augmented matrix form, we will perform operations on the rows that will lead us to the solution.<\/p><p id=\"fs-id1167829784535\">To solve by elimination, it doesn\u2019t matter which order we place the equations in the system. Similarly, in the matrix we can interchange the rows.<\/p><p id=\"fs-id1167836393210\">When we solve by elimination, we often multiply one of the equations by a constant. Since each row represents an equation, and we can multiply each side of an equation by a constant, similarly we can multiply each entry in a row by any real number except 0.<\/p><p>In elimination, we often add a multiple of one row to another row. In the matrix we can replace a row with its sum with a multiple of another row.<\/p><p id=\"fs-id1167836538695\">These actions are called row operations and will help us use the matrix to solve a system of equations.<\/p><div data-type=\"note\" id=\"fs-id1167833061563\"><div data-type=\"title\">Row Operations<\/div><p id=\"fs-id1167836456012\">In a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix.<\/p><ol id=\"fs-id1167829624876\" type=\"1\"><li>Interchange any two rows.<\/li><li>Multiply a row by any real number except 0.<\/li><li>Add a nonzero multiple of one row to another row.<\/li><\/ol><\/div><p id=\"fs-id1167836515322\">Performing these operations is easy to do but all the arithmetic can result in a mistake. If we use a system to record the row operation in each step, it is much easier to go back and check our work.<\/p><p id=\"fs-id1167824732306\">We use capital letters with subscripts to represent each row. We then show the operation to the left of the new matrix. To show interchanging a row:<\/p><span data-type=\"media\" id=\"fs-id1167829807711\" data-alt=\"A 2 by 3 matrix is shown. Its first row, labeled R2 is 2, minus 1, 2. Its second row, labeled R1 is 5, minus 3, minus 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_006_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A 2 by 3 matrix is shown. Its first row, labeled R2 is 2, minus 1, 2. Its second row, labeled R1 is 5, minus 3, minus 1.\"><\/span><p id=\"fs-id1167833129901\">To multiply row 2 by \\(-3\\):<\/p><span data-type=\"media\" id=\"fs-id1167836541327\" data-alt=\"A 2 by 3 matrix is shown. Its first row is 5, minus 3, minus 1. Its second row is 2, minus 1, 2. An arrow point from this matrix to another one on the right. The first row of the new matrix is the same. The second row is preceded by minus 3 R2. It is minus 6, 3, minus 6.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A 2 by 3 matrix is shown. Its first row is 5, minus 3, minus 1. Its second row is 2, minus 1, 2. An arrow point from this matrix to another one on the right. The first row of the new matrix is the same. The second row is preceded by minus 3 R2. It is minus 6, 3, minus 6.\"><\/span><p id=\"fs-id1167836558970\">To multiply row 2 by \\(-3\\) and add it to row 1:<\/p><span data-type=\"media\" id=\"fs-id1167829829286\" data-alt=\"A 2 by 3 matrix is shown. Its first row is 5, minus 3, minus 1. Its second row is 2, minus 1, 2. An arrow point from this matrix to another one on the right. The first row of the new matrix is preceded by minus 3 R2 plus R1. It is minus 1, 0, minus 7. The second row is 2, minus 1, 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_008_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A 2 by 3 matrix is shown. Its first row is 5, minus 3, minus 1. Its second row is 2, minus 1, 2. An arrow point from this matrix to another one on the right. The first row of the new matrix is preceded by minus 3 R2 plus R1. It is minus 1, 0, minus 7. The second row is 2, minus 1, 2.\"><\/span><div data-type=\"example\" id=\"fs-id1167833350263\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167824780787\"><div data-type=\"problem\" id=\"fs-id1167836499275\"><p id=\"fs-id1167826205547\">Perform the indicated operations on the augmented matrix:<\/p><p id=\"fs-id1167829748933\"><span class=\"token\">\u24d0<\/span> Interchange rows 2 and 3.<\/p><p id=\"fs-id1167833024294\"><span class=\"token\">\u24d1<\/span> Multiply row 2 by 5.<\/p><p id=\"fs-id1167829739413\"><span class=\"token\">\u24d2<\/span> Multiply row 3 by \\(-2\\) and add to row 1.<\/p><div data-type=\"equation\" id=\"fs-id1171792520714\" class=\"unnumbered\" data-label=\"\">\\(\\left[\\begin{array}{ccccccccc}6\\hfill &amp; &amp; &amp; \\hfill -5&amp; &amp; &amp; \\hfill 2&amp; &amp; \\\\ 2\\hfill &amp; &amp; &amp; \\hfill 1&amp; &amp; &amp; \\hfill -4&amp; &amp; \\\\ 3\\hfill &amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill 1&amp; &amp; \\end{array}|\\begin{array}{ccc}&amp; &amp; \\hfill 3\\\\ &amp; &amp; \\hfill 5\\\\ &amp; &amp; \\hfill -1\\end{array}\\right]\\)<\/div><\/div><div data-type=\"solution\" id=\"fs-id1167836645967\"><p id=\"fs-id1167833024483\"><span class=\"token\">\u24d0<\/span> We interchange rows 2 and 3.<\/p><div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167824669254\" data-alt=\"Two 3 by 4 matrices are shown. In the one on the left, the first row is 6, minus 5, 2, 3. The second row is 2, 1, minus 4, 5. The third row is 3, minus 3, 1, minus 1. The second matrix is similar except that rows 2 and 3 are interchanged.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Two 3 by 4 matrices are shown. In the one on the left, the first row is 6, minus 5, 2, 3. The second row is 2, 1, minus 4, 5. The third row is 3, minus 3, 1, minus 1. The second matrix is similar except that rows 2 and 3 are interchanged.\"><\/span><p id=\"fs-id1167836328953\"><span class=\"token\">\u24d1<\/span> We multiply row 2 by 5.<\/p><div data-type=\"newline\"><br><\/div> <span data-type=\"media\" data-alt=\"Two 3 by 4 matrices are shown. In the one on the left, the first row is 6, minus 5, 2, 3. The second row is 2, 1, minus 4, 5. The third row is 3, minus 3, 1, minus 1. The second matrix is similar to the first except that row 2, preceded by 5 R2, is 10, 5, minus 20, 25.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_010_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Two 3 by 4 matrices are shown. In the one on the left, the first row is 6, minus 5, 2, 3. The second row is 2, 1, minus 4, 5. The third row is 3, minus 3, 1, minus 1. The second matrix is similar to the first except that row 2, preceded by 5 R2, is 10, 5, minus 20, 25.\"><\/span><p id=\"fs-id1167829712000\"><span class=\"token\">\u24d2<\/span> We multiply row 3 by \\(-2\\) and add to row 1.<\/p><div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167836292453\" data-alt=\"In the 3 by 4 matrix, the first row is 6, minus 5, 2, 3. The second row is 2, 1, minus 4, 5. The third row is 3, minus 3, 1, minus 1. Performing the operation minus 2 R3 plus R1 on the first row, the first row becomes 6 plus minus 2 times 3, minus 5 plus minus 2 times minus 3, 2 plus minus 2 times 1 and 3 plus minus 2 times minus 1. This becomes 0, 1, 0, 5. The remaining 2 rows of the new matrix are the same.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_011_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In the 3 by 4 matrix, the first row is 6, minus 5, 2, 3. The second row is 2, 1, minus 4, 5. The third row is 3, minus 3, 1, minus 1. Performing the operation minus 2 R3 plus R1 on the first row, the first row becomes 6 plus minus 2 times 3, minus 5 plus minus 2 times minus 3, 2 plus minus 2 times 1 and 3 plus minus 2 times minus 1. This becomes 0, 1, 0, 5. The remaining 2 rows of the new matrix are the same.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829786590\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836335371\"><div data-type=\"problem\" id=\"fs-id1167829594338\"><p id=\"fs-id1167825993101\">Perform the indicated operations on the augmented matrix:<\/p><p id=\"fs-id1167829688134\"><span class=\"token\">\u24d0<\/span> Interchange rows 1 and 3.<\/p><p id=\"fs-id1167836334004\"><span class=\"token\">\u24d1<\/span> Multiply row 3 by 3.<\/p><p id=\"fs-id1167825691607\"><span class=\"token\">\u24d2<\/span> Multiply row 3 by 2 and add to row 2.<\/p><p id=\"fs-id1167833224470\">\\(\\left[\\begin{array}{ccccccccc}\\hfill 5&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill -2&amp; &amp; \\\\ \\hfill 4&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill -4&amp; &amp; \\\\ \\hfill -2&amp; &amp; &amp; \\hfill 3&amp; &amp; &amp; \\hfill 0&amp; &amp; \\end{array}|\\begin{array}{ccc}&amp; &amp; \\hfill -2\\\\ &amp; &amp; \\hfill 4\\\\ &amp; &amp; \\hfill -1\\end{array}\\right]\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167829627951\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left[\\begin{array}{cccccccccc}\\hfill -2&amp; &amp; &amp; \\hfill 3&amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill -2\\\\ \\hfill 4&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill -4&amp; &amp; &amp; \\hfill 4\\\\ \\hfill 5&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill -2\\end{array}\\right]\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left[\\begin{array}{cccccccccc}\\hfill -2&amp; &amp; &amp; \\hfill 3&amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill -2\\\\ \\hfill 4&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill -4&amp; &amp; &amp; \\hfill 4\\\\ 15\\hfill &amp; &amp; &amp; \\hfill -6&amp; &amp; &amp; \\hfill -6&amp; &amp; &amp; \\hfill -6\\end{array}\\right]\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\left[\\begin{array}{ccccccccccccc}\\hfill -2&amp; &amp; &amp; \\hfill 3&amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\\\ \\hfill 3&amp; &amp; &amp; \\hfill 4&amp; &amp; &amp; \\hfill -13&amp; &amp; &amp; \\hfill -16&amp; &amp; &amp; \\hfill -8\\\\ \\hfill 15&amp; &amp; &amp; \\hfill -6&amp; &amp; &amp; \\hfill -6&amp; &amp; &amp; \\hfill -6&amp; &amp; &amp; \\end{array}\\right]\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826172223\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836366482\"><div data-type=\"problem\" id=\"fs-id1167836732417\"><p id=\"fs-id1167836477875\">Perform the indicated operations on the augmented matrix:<\/p><p id=\"fs-id1167836300769\"><span class=\"token\">\u24d0<\/span> Interchange rows 1 and 2,<\/p><p id=\"fs-id1167836611076\"><span class=\"token\">\u24d1<\/span> Multiply row 1 by 2,<\/p><p id=\"fs-id1167829859334\"><span class=\"token\">\u24d2<\/span> Multiply row 2 by 3 and add to row 1.<\/p><p id=\"fs-id1167836352756\">\\(\\left[\\begin{array}{ccccccccc}2\\hfill &amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill -2&amp; &amp; \\\\ 4\\hfill &amp; &amp; &amp; \\hfill 1&amp; &amp; &amp; \\hfill -3&amp; &amp; \\\\ 5\\hfill &amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill 4&amp; &amp; \\end{array}|\\begin{array}{ccc}&amp; &amp; \\hfill -4\\\\ &amp; &amp; \\hfill 2\\\\ &amp; &amp; \\hfill -1\\end{array}\\right]\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829841016\"><p><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left[\\begin{array}{cccccccccc}4\\hfill &amp; &amp; &amp; \\hfill 1&amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill 2\\\\ 2\\hfill &amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill -4\\\\ 5\\hfill &amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill 4&amp; &amp; &amp; \\hfill -1\\end{array}\\right]\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left[\\begin{array}{cccccccccc}8\\hfill &amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill -6&amp; &amp; &amp; \\hfill 4\\\\ 2\\hfill &amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill -4\\\\ 5\\hfill &amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill 4&amp; &amp; &amp; \\hfill -1\\end{array}\\right]\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\left[\\begin{array}{cccccccccc}14\\hfill &amp; &amp; &amp; \\hfill -7&amp; &amp; &amp; \\hfill -12&amp; &amp; &amp; \\hfill -8\\\\ 2\\hfill &amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill -4\\\\ 5\\hfill &amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill 4&amp; &amp; &amp; \\hfill -1\\end{array}\\right]\\)<\/div><\/div><\/div><p id=\"fs-id1167836515290\">Now that we have practiced the row operations, we will look at an augmented matrix and figure out what operation we will use to reach a goal. This is exactly what we did when we did elimination. We decided what number to multiply a row by in order that a variable would be eliminated when we added the rows together.<\/p><p id=\"fs-id1167833377541\">Given this system, what would you do to eliminate <em data-effect=\"italics\">x<\/em>?<\/p><span data-type=\"media\" id=\"fs-id1167829786049\" data-alt=\"The two equations are x minus y equals 2 and 4x minus 8y equals 0. Multiplying the first by minus 4, we get minus 4x plus 4y equals minus 8. Adding this to the second equation we get minus 4y equals minus 8.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_012_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The two equations are x minus y equals 2 and 4x minus 8y equals 0. Multiplying the first by minus 4, we get minus 4x plus 4y equals minus 8. Adding this to the second equation we get minus 4y equals minus 8.\"><\/span><p id=\"fs-id1167836540507\">This next example essentially does the same thing, but to the matrix.<\/p><div data-type=\"example\" id=\"fs-id1167836409774\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836319795\"><p id=\"fs-id1167836768391\">Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix: \\(\\left[\\begin{array}{cccccc}1\\hfill &amp; &amp; &amp; \\hfill -1&amp; &amp; \\\\ 4\\hfill &amp; &amp; &amp; \\hfill -8&amp; &amp; \\end{array}|\\begin{array}{ccc}&amp; &amp; \\hfill 2\\\\ &amp; &amp; \\hfill 0\\end{array}\\right].\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167829624231\">To make the 4 a 0, we could multiply row 1 by \\(-4\\) and then add it to row 2.<\/p><span data-type=\"media\" id=\"fs-id1167833365764\" data-alt=\"The 2 by 3 matrix is 1, minus 1, 2 and 4, minus 8, 0. Performing the operation minus 4R1 plus R2 on row 2, the second row of the new matrix becomes 0, minus 4, minus 8. The first row remains the same.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_013_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The 2 by 3 matrix is 1, minus 1, 2 and 4, minus 8, 0. Performing the operation minus 4R1 plus R2 on row 2, the second row of the new matrix becomes 0, minus 4, minus 8. The first row remains the same.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836524340\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836630643\"><div data-type=\"problem\" id=\"fs-id1167833279820\"><p id=\"fs-id1167833196797\">Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix: \\(\\left[\\begin{array}{cccccc}1\\hfill &amp; &amp; &amp; \\hfill -1&amp; &amp; \\\\ 3\\hfill &amp; &amp; &amp; \\hfill -6&amp; &amp; \\end{array}|\\begin{array}{ccc}&amp; &amp; \\hfill 2\\\\ &amp; &amp; \\hfill 2\\end{array}\\right].\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833310074\"><p id=\"fs-id1167833025573\">\\(\\left[\\begin{array}{ccccccc}1\\hfill &amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill 2\\\\ 0\\hfill &amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill -4\\end{array}\\right]\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836531933\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829831866\"><div data-type=\"problem\" id=\"fs-id1167833202534\"><p id=\"fs-id1167829586303\">Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix: \\(\\left[\\begin{array}{cccccc}\\hfill 1&amp; &amp; &amp; \\hfill -1&amp; &amp; \\\\ \\hfill -2&amp; &amp; &amp; \\hfill -3&amp; &amp; \\end{array}|\\begin{array}{ccc}&amp; &amp; \\hfill 3\\\\ &amp; &amp; \\hfill 2\\end{array}\\right].\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836535460\"><p>\\(\\left[\\begin{array}{ccccccc}1\\hfill &amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill 3\\\\ 0\\hfill &amp; &amp; &amp; \\hfill -5&amp; &amp; &amp; \\hfill 8\\end{array}\\right]\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\"><h3 data-type=\"title\">Solve Systems of Equations Using Matrices<\/h3><p id=\"fs-id1167833338960\">To solve a system of equations using matrices, we transform the augmented matrix into a matrix in <span data-type=\"term\">row-echelon form<\/span> using row operations. For a consistent and independent system of equations, its <span data-type=\"term\" class=\"no-emphasis\">augmented matrix<\/span> is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.<\/p><div data-type=\"note\" id=\"fs-id1167836363886\"><div data-type=\"title\">Row-Echelon Form<\/div><p id=\"fs-id1167829578768\">For a consistent and independent system of equations, its augmented matrix is in <strong data-effect=\"bold\">row-echelon form<\/strong> when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.<\/p><span data-type=\"media\" data-alt=\"A 2 by 3 matrix is shown on the left. Its first row is 1, a, b. Its second row is 0, 1, c. An arrow points diagonally down and right, overlapping both the 1s in the matrix. A 3 by 4 matrix is shown on the right. Its first row is 1, a, b, d. Its second row is 0, 1, c, e. Its third row is 0, 0, 1, f. An arrow points diagonally down and right, overlapping all the 1s in the matrix. a, b, c, d, e, f are real numbers.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_014_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A 2 by 3 matrix is shown on the left. Its first row is 1, a, b. Its second row is 0, 1, c. An arrow points diagonally down and right, overlapping both the 1s in the matrix. A 3 by 4 matrix is shown on the right. Its first row is 1, a, b, d. Its second row is 0, 1, c, e. Its third row is 0, 0, 1, f. An arrow points diagonally down and right, overlapping all the 1s in the matrix. a, b, c, d, e, f are real numbers.\"><\/span><\/div><p>Once we get the augmented matrix into row-echelon form, we can write the equivalent system of equations and read the value of at least one variable. We then substitute this value in another equation to continue to solve for the other variables. This process is illustrated in the next example.<\/p><div data-type=\"example\" id=\"fs-id1167829692902\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Solve a System of Equations Using a Matrix<\/div><div data-type=\"exercise\" id=\"fs-id1167829743345\"><div data-type=\"problem\" id=\"fs-id1167822916232\"><p id=\"fs-id1167830093682\">Solve the system of equations using a matrix: \\(\\left\\{\\begin{array}{c}3x+4y=5\\hfill \\\\ x+2y=1\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167822996975\"><span data-type=\"media\" id=\"fs-id1167836525799\" data-alt=\"The equations are 3x plus 4y equals 5 and x plus 2y equals 1. Step 1. Write the augmented matrix for the system of equations. We get a 2 by 3 matrix with first row 3, 4, 5 and second row 1, 2, 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_015a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are 3x plus 4y equals 5 and x plus 2y equals 1. Step 1. Write the augmented matrix for the system of equations. We get a 2 by 3 matrix with first row 3, 4, 5 and second row 1, 2, 1.\"><\/span><span data-type=\"media\" id=\"fs-id1167829828529\" data-alt=\"Step 2. Using row operations get the entry in row 1, column 1 to be 1. Interchange rows R1 and R2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_015b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2. Using row operations get the entry in row 1, column 1 to be 1. Interchange rows R1 and R2.\"><\/span><span data-type=\"media\" id=\"fs-id1167833382010\" data-alt=\"Step 3. Using row operations, get zeros in column 1 below the 1. Multiply row 1 by minus 3 and add it to row 2. Row 2 becomes 0, minus 2, 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_015c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3. Using row operations, get zeros in column 1 below the 1. Multiply row 1 by minus 3 and add it to row 2. Row 2 becomes 0, minus 2, 2.\"><\/span><span data-type=\"media\" id=\"fs-id1167836692585\" data-alt=\"Step 4. Using row operations, get the entry in row 2, column 2 to be 1. Multiply row 2 by minus half. Row 2 becomes 0, 1, minus 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_015d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4. Using row operations, get the entry in row 2, column 2 to be 1. Multiply row 2 by minus half. Row 2 becomes 0, 1, minus 1.\"><\/span><span data-type=\"media\" id=\"fs-id1167833021336\" data-alt=\"Step 5. Continue the process until the matrix is in row-echelon form. The matrix is now in row-echelon form.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_015e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5. Continue the process until the matrix is in row-echelon form. The matrix is now in row-echelon form.\"><\/span><span data-type=\"media\" id=\"fs-id1167836713900\" data-alt=\"Step 6. Write the corresponding system of equations. We get x plus 2y equals 1 and y equals minus 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_015f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 6. Write the corresponding system of equations. We get x plus 2y equals 1 and y equals minus 1.\"><\/span><span data-type=\"media\" id=\"fs-id1167836688662\" data-alt=\"Step 7. Use substitution to find the remaining variables. Substitute y equals negative 1 into x plus 2y equals 1. X plus 2 times negative 1 equals 1. X minus 2 equals 1. We get x equal to 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_015g_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 7. Use substitution to find the remaining variables. Substitute y equals negative 1 into x plus 2y equals 1. X plus 2 times negative 1 equals 1. X minus 2 equals 1. We get x equal to 3.\"><\/span><span data-type=\"media\" id=\"fs-id1167832930182\" data-alt=\"Step 8. Write the solution as an ordered pair or triple. Ordered pair is (3, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_015h_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 8. Write the solution as an ordered pair or triple. Ordered pair is (3, negative 1).\"><\/span><span data-type=\"media\" id=\"fs-id1167836542466\" data-alt=\"Step 9. Check that the solution makes the original equations true.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_015i_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 9. Check that the solution makes the original equations true.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167836484478\">Solve the system of equations using a matrix: \\(\\left\\{\\begin{array}{c}2x+y=7\\hfill \\\\ x-2y=6\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836533019\"><p id=\"fs-id1167825823995\">The solution is \\(\\left(4,-1\\right).\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836621323\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836518516\"><div data-type=\"problem\" id=\"fs-id1167836575707\"><p id=\"fs-id1167824852264\">Solve the system of equations using a matrix: \\(\\left\\{\\begin{array}{c}2x+y=-4\\hfill \\\\ x-y=-2\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167836552453\">The solution is \\(\\left(-2,0\\right).\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167833227319\">The steps are summarized here.<\/p><div data-type=\"note\" id=\"fs-id1167833023848\" class=\"howto\"><div data-type=\"title\">Solve a system of equations using matrices.<\/div><ol id=\"fs-id1167836485160\" type=\"1\" class=\"stepwise\"><li>Write the augmented matrix for the system of equations.<\/li><li>Using row operations get the entry in row 1, column 1 to be 1.<\/li><li>Using row operations, get zeros in column 1 below the 1.<\/li><li>Using row operations, get the entry in row 2, column 2 to be 1.<\/li><li>Continue the process until the matrix is in row-echelon form.<\/li><li>Write the corresponding system of equations.<\/li><li>Use substitution to find the remaining variables.<\/li><li>Write the solution as an ordered pair or triple.<\/li><li>Check that the solution makes the original equations true.<\/li><\/ol><\/div><p id=\"fs-id1167833058881\">Here is a visual to show the order for getting the 1\u2019s and 0\u2019s in the proper position for row-echelon form.<\/p><span data-type=\"media\" id=\"fs-id1167836295125\" data-alt=\"The figure shows 3 steps for a 2 by 3 matrix and 6 steps for a 3 by 4 matrix. For the former, step 1 is to get a 1 in row 1 column 1. Step to is to get a 0 is row 2 column 1. Step 3 is to get a 1 in row 2 column 2. For a 3 by 4 matrix, step 1 is to get a 1 in row 1 column 1. Step 2 is to get a 0 in row 2 column 1. Step 3 is to get a 0 in row 3 column 1. Step 4 is to get a 1 in row 2 column 2. Step 5 is to get a 0 in row 3 column 2. Step 6 is to get a 1 in row 3 column 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_016_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows 3 steps for a 2 by 3 matrix and 6 steps for a 3 by 4 matrix. For the former, step 1 is to get a 1 in row 1 column 1. Step to is to get a 0 is row 2 column 1. Step 3 is to get a 1 in row 2 column 2. For a 3 by 4 matrix, step 1 is to get a 1 in row 1 column 1. Step 2 is to get a 0 in row 2 column 1. Step 3 is to get a 0 in row 3 column 1. Step 4 is to get a 1 in row 2 column 2. Step 5 is to get a 0 in row 3 column 2. Step 6 is to get a 1 in row 3 column 3.\"><\/span><p id=\"fs-id1168757403507\">We use the same procedure when the system of equations has three equations.<\/p><div data-type=\"example\" id=\"fs-id1167832984010\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829812225\"><div data-type=\"problem\" id=\"fs-id1167833378931\"><p id=\"fs-id1167829690073\">Solve the system of equations using a matrix: \\(\\left\\{\\begin{array}{c}3x+8y+2z=-5\\hfill \\\\ 2x+5y-3z=0\\hfill \\\\ x+2y-2z=-1\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\"><table id=\"fs-id1167829714495\" class=\"unnumbered unstyled can-break\" summary=\"The equations are 3x plus 8y plus 2z equals minus 5, 2x plus 5y minus 3z equals 0, x plus 2y minus 2z equals minus 1. Write the augmented matrix for the equations. Row 1 is 3, 8, 2, minus 5. Row 2 is 2, 5, minus 3, 0. Row 3 is 1, 2, minus 2, minus 1. Interchange row 1 and 3 to get the entry in row 1, column 1 to be 1. Use operation minus 2R1 plus R2 on row 2. Use operation minus 3R1 plus R3 on row 3. Use operation minus 2R2 plus R3 on row 3. Use operation 1 upon 6 R3 on row 3. The matrix is now in row-echelon form. The corresponding system of equations is x plus 2y minus 2z equals minus 1, y plus z equals 2 and z equals minus 1. Using substitution, we get y equal to 3 and x equal to minus 9. The solution is minus 9, 3, minus 1. Check that the original equations hold true.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836309310\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the augmented matrix for the equations.<\/td><td 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_05_017b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Interchange row 1 and 3 to get the entry in<div data-type=\"newline\"><br><\/div>row 1, column 1 to be 1.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836329209\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Using row operations, get zeros in column 1 below the 1.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836531979\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836340874\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The entry in row 2, column 2 is now 1.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Continue the process until the matrix<div data-type=\"newline\"><br><\/div>is in row-echelon form.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167823026715\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829748610\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The matrix is now in row-echelon form.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829744236\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the corresponding system of equations.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824852333\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use substitution to find the remaining variables.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824740544\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829624548\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the solution as an ordered pair or triple.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836477846\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017n_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check that the solution makes the original equations true.<\/td><td data-valign=\"top\" data-align=\"left\">We leave the check for you.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829807014\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836548820\"><div data-type=\"problem\" id=\"fs-id1167833380079\"><p>Solve the system of equations using a matrix: \\(\\left\\{\\begin{array}{c}2x-5y+3z=8\\hfill \\\\ 3x-y+4z=7\\hfill \\\\ x+3y+2z=-3\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833020740\"><p id=\"fs-id1167829709377\">\\(\\left(6,-1,-3\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833009146\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836685150\"><div data-type=\"problem\" id=\"fs-id1167836388313\"><p id=\"fs-id1167836330398\">Solve the system of equations using a matrix: \\(\\left\\{\\begin{array}{c}\\hfill -3x+y+z=-4\\\\ \\hfill \\text{\u2212}x+2y-2z=1\\\\ 2x-y-z=-1\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836299894\"><p id=\"fs-id1167833024271\">\\(\\left(5,7,4\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167833158290\">So far our work with matrices has only been with systems that are consistent and independent, which means they have exactly one solution. Let\u2019s now look at what happens when we use a matrix for a dependent or inconsistent system.<\/p><div data-type=\"example\" id=\"fs-id1167836693828\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836357145\"><div data-type=\"problem\" id=\"fs-id1167836697732\"><p id=\"fs-id1167829688774\">Solve the system of equations using a matrix: \\(\\left\\{\\begin{array}{c}x+y+3z=0\\hfill \\\\ x+3y+5z=0\\hfill \\\\ 2x+4z=1\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826172431\"><table id=\"fs-id1167836320619\" class=\"unnumbered unstyled can-break\" summary=\"The equations are x plus y plus 3z equals 0, x plus 3y plus 5z equals 0 and 2x plus 4z equals 1. The first row of the augmented matrix is 1, 1, 3, 0. Row 2 is 1, 3, 5, 0. Row 3 is 2, 0, 4, 1. Use row operation minus 1R1 plus R2 on row 2. Use operation minus 2R1 plus R3 on row 3. Use operation half R2 on row 2. Use operation 2R2 plus R3. The corresponding equations are x plus y plus 3z equals 0, y plus z equals 0 and 0 not equal to 1.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829693728\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_018a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the augmented matrix for the equations.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836552968\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_018b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The entry in row 1, column 1 is 1.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Using row operations, get zeros in column 1 below the 1.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836697832\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_018c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836628663\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_018d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Continue the process until the matrix is in row-echelon form.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836339886\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_018e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply row 2 by 2 and add it to row 3.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836605395\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_018f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">At this point, we have all zeros on the left of row 3.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the corresponding system of equations.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829597744\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_018g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Since \\(0\\ne 1\\) we have a false statement. Just as when we solved a system using other methods, this tells us we have an inconsistent system. There is no solution.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836507507\"><div data-type=\"problem\"><p id=\"fs-id1167836656607\">Solve the system of equations using a matrix: \\(\\left\\{\\begin{array}{c}x-2y+2z=1\\hfill \\\\ \\hfill -2x+y-z=2\\\\ x-y+z=5\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836697856\"><p id=\"fs-id1167836556734\">no solution<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836597057\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836699123\"><div data-type=\"problem\"><p id=\"fs-id1167836311815\">Solve the system of equations using a matrix: \\(\\left\\{\\begin{array}{c}3x+4y-3z=-2\\hfill \\\\ 2x+3y-z=-12\\hfill \\\\ x+y-2z=6\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836585335\"><p id=\"fs-id1167829686514\">no solution<\/p><\/div><\/div><\/div><p id=\"fs-id1167833019836\">The last system was inconsistent and so had no solutions. The next example is dependent and has infinitely many solutions.<\/p><div data-type=\"example\" id=\"fs-id1167836555721\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167832925613\"><div data-type=\"problem\" id=\"fs-id1167836535076\"><p id=\"fs-id1167829807672\">Solve the system of equations using a matrix: \\(\\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-id1167836484491\"><table id=\"fs-id1167836701231\" class=\"unnumbered unstyled can-break\" summary=\"The equations are x minus 2y plus 3z equals 1, x plus y minus 3z equals 7 and 3x minus 4y plus 5z equals 7. The augmented matrix is: row 1: 1, minus 2, 3, 1, row 2: 1, 1, minus 3, 7, row 3: 3, minus 4, 5, 7. Use operation minus 1R1 plus R2 on row 2. use operation minus 3 R1 plus R3 on row 3. Use operation 1 upon 3 R2 on row 2. Use operation minus 2R2 plus R3 on row 3. The corresponding equations are x minus 2y plus 3z is 1, y minus 2z is 2 and 0 is 0. Since 0 is 0 we have a true statement. Just as when we solved by substitution, this tells us we have a dependent system. There are infinitely many solutions. Solving for y in second equation, we get y equal to 2z plus 2. Substituting this in the first equation, we get x equal to z plus 5. The system has infinitely many solutions x, y, z where x is z plus 5 and y is 2z plus 2 and z is any real number.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836662838\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the augmented matrix for the equations.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836352824\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The entry in row 1, column 1 is 1.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Using row operations, get zeros in column 1 below the 1.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836529008\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td 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_05_019d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Continue the process until the matrix is in row-echelon form.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833256119\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply row 2 by \\(-2\\) and add it to row 3.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829741959\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">At this point, we have all zeros in the bottom row.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the corresponding system of equations.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833245758\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Since \\(0=0\\) we have a true statement. Just as when we solved by substitution, this tells us we have a dependent system. There are infinitely many solutions.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">y<\/em> in terms of <em data-effect=\"italics\">z<\/em> in the second equation.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836493258\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve the first equation for <em data-effect=\"italics\">x<\/em> in terms of <em data-effect=\"italics\">z<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836707050\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute \\(y=2z+2.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829720122\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829721214\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836649335\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829720944\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">The system has infinitely many solutions \\(\\left(x,y,z\\right)\\text{,}\\) where\\(x=z+5;y=2z+2;z\\) is any real number.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833207815\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829692179\"><div data-type=\"problem\" id=\"fs-id1167836600243\"><p id=\"fs-id1167829745733\">Solve the system of equations using a matrix: \\(\\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-id1167836293696\"><p>infinitely many solutions \\(\\left(x,y,z\\right)\\text{,}\\) where \\(x=z-3;y=3;z\\) is any real number.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836315181\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836349086\"><div data-type=\"problem\" id=\"fs-id1167836310628\"><p id=\"fs-id1167833158281\">Solve the system of equations using a matrix: \\(\\left\\{\\begin{array}{c}x-y-z=1\\hfill \\\\ \\hfill \\text{\u2212}x+2y-3z=-4\\\\ 3x-2y-7z=0\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829843556\"><p id=\"fs-id1167833059149\">infinitely many solutions \\(\\left(x,y,z\\right)\\text{,}\\) where \\(x=5z-2;y=4z-3;z\\) is any real number.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829748849\" class=\"media-2\"><p id=\"fs-id1167833274200\">Access this online resource for additional instruction and practice with Gaussian Elimination.<\/p><ul id=\"fs-id1169145728353\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37GaussElim\">Gaussian Elimination<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836626308\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167836293453\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Matrix:<\/strong> A matrix is a rectangular array of numbers arranged in rows and columns. A matrix with <em data-effect=\"italics\">m<\/em> rows and <em data-effect=\"italics\">n<\/em> columns has <em data-effect=\"italics\">order<\/em> \\(m\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}n.\\) The matrix on the left below has 2 rows and 3 columns and so it has order \\(2\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3.\\) We say it is a 2 by 3 matrix.<div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167833385766\" data-alt=\"Figure shows two matrices. The one on the left has the numbers minus 3, minus 2 and 2 in the first row and the numbers minus 1, 4 and 5 in the second row. The rows and columns are enclosed within brackets. Thus, it has 2 rows and 3 columns. It is labeled 2 cross 3 or 2 by 3 matrix. The matrix on the right is similar but with 3 rows and 4 columns. It is labeled 3 by 4 matrix.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_020_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows two matrices. The one on the left has the numbers minus 3, minus 2 and 2 in the first row and the numbers minus 1, 4 and 5 in the second row. The rows and columns are enclosed within brackets. Thus, it has 2 rows and 3 columns. It is labeled 2 cross 3 or 2 by 3 matrix. The matrix on the right is similar but with 3 rows and 4 columns. It is labeled 3 by 4 matrix.\"><\/span><div data-type=\"newline\"><br><\/div> Each number in the matrix is called an <em data-effect=\"italics\">element<\/em> or <em data-effect=\"italics\">entry<\/em> in the matrix.<\/li><li><strong data-effect=\"bold\">Row Operations:<\/strong> In a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix. <ul id=\"fs-id1167836613366\" data-bullet-style=\"bullet\"><li>Interchange any two rows<\/li><li>Multiply a row by any real number except 0<\/li><li>Add a nonzero multiple of one row to another row<\/li><\/ul><\/li><li><strong data-effect=\"bold\">Row-Echelon Form:<\/strong> For a consistent and independent system of equations, its augmented matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.<div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167833316687\" data-alt=\"Figure shows two matrices. The one on the left has the numbers minus 3, minus 2 and 2 in the first row and the numbers minus 1, 4 and 5 in the second row. The rows and columns are enclosed within brackets. Thus, it has 2 rows and 3 columns. It is labeled 2 cross 3 or 2 by 3 matrix. The matrix on the right is similar but with 3 rows and 4 columns. It is labeled 3 by 4 matrix.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_021_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows two matrices. The one on the left has the numbers minus 3, minus 2 and 2 in the first row and the numbers minus 1, 4 and 5 in the second row. The rows and columns are enclosed within brackets. Thus, it has 2 rows and 3 columns. It is labeled 2 cross 3 or 2 by 3 matrix. The matrix on the right is similar but with 3 rows and 4 columns. It is labeled 3 by 4 matrix.\"><\/span> <\/li><li><strong data-effect=\"bold\">How to solve a system of equations using matrices.<\/strong><ol id=\"fs-id1167836386945\" type=\"1\" class=\"stepwise\"><li>Write the augmented matrix for the system of equations.<\/li><li>Using row operations get the entry in row 1, column 1 to be 1.<\/li><li>Using row operations, get zeros in column 1 below the 1.<\/li><li>Using row operations, get the entry in row 2, column 2 to be 1.<\/li><li>Continue the process until the matrix is in row-echelon form.<\/li><li>Write the corresponding system of equations.<\/li><li>Use substitution to find the remaining variables.<\/li><li>Write the solution as an ordered pair or triple.<\/li><li>Check that the solution makes the original equations true.<\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167833025558\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167836449701\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167836547542\"><strong data-effect=\"bold\">Write the Augmented Matrix for a System of Equations<\/strong><\/p><p id=\"fs-id1167826171377\">In the following exercises, write each system of linear equations as an augmented matrix.<\/p><div data-type=\"exercise\" id=\"fs-id1167836523331\"><div data-type=\"problem\" id=\"fs-id1167829844031\"><p id=\"fs-id1167836620562\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}3x-y=-1\\hfill \\\\ 2y=2x+5\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left\\{\\begin{array}{c}4x+3y=-2\\hfill \\\\ x-2y-3z=7\\hfill \\\\ 2x-y+2z=-6\\hfill \\end{array}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836502030\"><div data-type=\"problem\" id=\"fs-id1167833338855\"><p id=\"fs-id1167836579407\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}2x+4y=-5\\hfill \\\\ 3x-2y=2\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left\\{\\begin{array}{c}3x-2y-z=-2\\hfill \\\\ -2x+y=5\\hfill \\\\ 5x+4y+z=-1\\hfill \\end{array}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167824781184\"><p><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left[\\begin{array}{ccccccc}2\\hfill &amp; &amp; &amp; \\hfill 4&amp; &amp; &amp; \\hfill -5\\\\ 3\\hfill &amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill 2\\end{array}\\right]\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left[\\begin{array}{cccccccccc}\\hfill 3&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill -2\\\\ \\hfill -2&amp; &amp; &amp; \\hfill 1&amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill 5\\\\ \\hfill 5&amp; &amp; &amp; \\hfill 4&amp; &amp; &amp; \\hfill 1&amp; &amp; &amp; \\hfill -1\\end{array}\\right]\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829739049\"><div data-type=\"problem\" id=\"fs-id1167829714126\"><p id=\"fs-id1167833361725\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}3x-y=-4\\hfill \\\\ 2x=y+2\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left\\{\\begin{array}{c}x-3y-4z=-2\\hfill \\\\ 4x+2y+2z=5\\hfill \\\\ 2x-5y+7z=-8\\hfill \\end{array}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833009852\"><div data-type=\"problem\" id=\"fs-id1167829598294\"><p id=\"fs-id1167836364054\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}2x-5y=-3\\hfill \\\\ 4x=3y-1\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left\\{\\begin{array}{c}4x+3y-2z=-3\\hfill \\\\ \\hfill -2x+y-3z=4\\\\ \\hfill \\text{\u2212}x-4y+5z=-2\\end{array}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167836607370\"><p id=\"fs-id1167825830228\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left[\\begin{array}{ccccccc}2\\hfill &amp; &amp; &amp; \\hfill -5&amp; &amp; &amp; \\hfill -3\\\\ 4\\hfill &amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill -1\\end{array}\\right]\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left[\\begin{array}{cccccccccc}4\\hfill &amp; &amp; &amp; \\hfill 3&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill -3\\\\ \\hfill -2&amp; &amp; &amp; \\hfill 1&amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill 4\\\\ -1\\hfill &amp; &amp; &amp; \\hfill -4&amp; &amp; &amp; \\hfill 5&amp; &amp; &amp; \\hfill -2\\end{array}\\right]\\)<\/div><\/div><p id=\"fs-id1167833082484\">Write the system of equations that corresponds to the augmented matrix.<\/p><div data-type=\"exercise\" id=\"fs-id1167836391699\"><div data-type=\"problem\" id=\"fs-id1167829788432\"><p id=\"fs-id1167829787229\">\\(\\left[\\begin{array}{cccc}2\\hfill &amp; &amp; &amp; \\hfill -1\\\\ 1\\hfill &amp; &amp; &amp; \\hfill -3\\end{array}\\phantom{\\rule{0.5em}{0ex}}|\\phantom{\\rule{0.5em}{0ex}}\\begin{array}{c}\\hfill 4\\\\ \\hfill 2\\end{array}\\right]\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829692032\"><div data-type=\"problem\"><p id=\"fs-id1167836706030\">\\(\\left[\\begin{array}{cccc}2\\hfill &amp; &amp; &amp; \\hfill -4\\\\ 3\\hfill &amp; &amp; &amp; \\hfill -3\\end{array}\\phantom{\\rule{0.5em}{0ex}}|\\phantom{\\rule{0.5em}{0ex}}\\begin{array}{c}\\hfill -2\\\\ \\hfill -1\\end{array}\\right]\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836417217\"><p id=\"fs-id1167836542591\">\\(\\left\\{\\begin{array}{c}2x-4y=-2\\hfill \\\\ 3x-3y=-1\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836448155\"><div data-type=\"problem\" id=\"fs-id1167833207785\"><p id=\"fs-id1167836626772\">\\(\\left[\\begin{array}{ccccccc}1\\hfill &amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill -3\\\\ 1\\hfill &amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill 0\\\\ 0\\hfill &amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill 2\\end{array}\\phantom{\\rule{0.5em}{0ex}}|\\phantom{\\rule{0.5em}{0ex}}\\begin{array}{c}\\hfill -1\\\\ \\hfill -2\\\\ \\hfill 3\\end{array}\\right]\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829832302\"><div data-type=\"problem\" id=\"fs-id1167836732801\"><p>\\(\\left[\\begin{array}{ccccccc}2\\hfill &amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill 0\\\\ 0\\hfill &amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill -1\\\\ 3\\hfill &amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill -1\\end{array}\\phantom{\\rule{0.5em}{0ex}}|\\phantom{\\rule{0.5em}{0ex}}\\begin{array}{c}\\hfill -1\\\\ \\hfill 2\\\\ \\hfill -2\\end{array}\\right]\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836722460\"><p id=\"fs-id1167836627860\">\\(\\left\\{\\begin{array}{c}2x-2y=-1\\hfill \\\\ 2y-z=2\\hfill \\\\ 3x-z=-2\\hfill \\end{array}\\)<\/p><\/div><\/div><p id=\"fs-id1167829688535\"><strong data-effect=\"bold\">Use Row Operations on a Matrix<\/strong><\/p><p id=\"fs-id1167836613644\">In the following exercises, perform the indicated operations on the augmented matrices.<\/p><div data-type=\"exercise\" id=\"fs-id1167836622536\"><div data-type=\"problem\" id=\"fs-id1167836287993\"><p id=\"fs-id1167836367083\">\\(\\left[\\begin{array}{cccc}6\\hfill &amp; &amp; &amp; \\hfill -4\\\\ 3\\hfill &amp; &amp; &amp; \\hfill -2\\end{array}\\phantom{\\rule{0.5em}{0ex}}|\\phantom{\\rule{0.5em}{0ex}}\\begin{array}{c}\\hfill 3\\\\ \\hfill 1\\end{array}\\right]\\)<\/p><p id=\"fs-id1167829690924\"><span class=\"token\">\u24d0<\/span> Interchange rows 1 and 2<\/p><p id=\"fs-id1167836296936\"><span class=\"token\">\u24d1<\/span> Multiply row 2 by 3<\/p><p id=\"fs-id1167836540243\"><span class=\"token\">\u24d2<\/span> Multiply row 2 by \\(-2\\) and add row 1 to it.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836538567\"><div data-type=\"problem\" id=\"fs-id1167829696382\"><p id=\"fs-id1167833014865\">\\(\\left[\\begin{array}{cccc}4\\hfill &amp; &amp; &amp; \\hfill -6\\\\ 3\\hfill &amp; &amp; &amp; \\hfill 2\\end{array}\\phantom{\\rule{0.5em}{0ex}}|\\phantom{\\rule{0.5em}{0ex}}\\begin{array}{c}\\hfill -3\\\\ \\hfill 1\\end{array}\\right]\\)<\/p><p id=\"fs-id1167836650085\"><span class=\"token\">\u24d0<\/span> Interchange rows 1 and 2<\/p><p id=\"fs-id1167836627418\"><span class=\"token\">\u24d1<\/span> Multiply row 1 by 4<\/p><p id=\"fs-id1167836362828\"><span class=\"token\">\u24d2<\/span> Multiply row 2 by 3 and add row 1 to it.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836418067\"><p id=\"fs-id1167833054679\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left[\\begin{array}{ccccccc}3\\hfill &amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill 1\\\\ 4\\hfill &amp; &amp; &amp; \\hfill -6&amp; &amp; &amp; \\hfill -3\\end{array}\\right]\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left[\\begin{array}{ccccccc}12\\hfill &amp; &amp; &amp; \\hfill 8&amp; &amp; &amp; \\hfill 4\\\\ 4\\hfill &amp; &amp; &amp; \\hfill -6&amp; &amp; &amp; \\hfill -3\\end{array}\\right]\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\left[\\begin{array}{ccccccc}12\\hfill &amp; &amp; &amp; \\hfill 8&amp; &amp; &amp; \\hfill 4\\\\ 24\\hfill &amp; &amp; &amp; \\hfill -10&amp; &amp; &amp; \\hfill -5\\end{array}\\right]\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836706572\"><div data-type=\"problem\" id=\"fs-id1167829695986\"><p id=\"fs-id1167836625455\">\\(\\left[\\begin{array}{ccccccc}4\\hfill &amp; &amp; &amp; \\hfill -12&amp; &amp; &amp; \\hfill -8\\\\ 4\\hfill &amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill -3\\\\ -6\\hfill &amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill -1\\end{array}\\phantom{\\rule{0.5em}{0ex}}|\\phantom{\\rule{0.5em}{0ex}}\\begin{array}{c}\\hfill 16\\\\ \\hfill -1\\\\ \\hfill -1\\end{array}\\right]\\)<\/p><p id=\"fs-id1167829620875\"><span class=\"token\">\u24d0<\/span> Interchange rows 2 and 3<\/p><p><span class=\"token\">\u24d1<\/span> Multiply row 1 by 4<\/p><p id=\"fs-id1167833361899\"><span class=\"token\">\u24d2<\/span> Multiply row 2 by \\(-2\\) and add to row 3.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829811863\"><div data-type=\"problem\" id=\"fs-id1167829752192\"><p id=\"fs-id1167833380181\">\\(\\left[\\begin{array}{ccccccc}6\\hfill &amp; &amp; &amp; \\hfill -5&amp; &amp; &amp; \\hfill 2\\\\ 2\\hfill &amp; &amp; &amp; \\hfill 1&amp; &amp; &amp; \\hfill -4\\\\ 3\\hfill &amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill 1\\end{array}\\phantom{\\rule{0.5em}{0ex}}|\\phantom{\\rule{0.5em}{0ex}}\\begin{array}{c}\\hfill 3\\\\ \\hfill 5\\\\ \\hfill -1\\end{array}\\right]\\)<\/p><p id=\"fs-id1167829579826\"><span class=\"token\">\u24d0<\/span> Interchange rows 2 and 3<\/p><p id=\"fs-id1167830077562\"><span class=\"token\">\u24d1<\/span> Multiply row 2 by 5<\/p><p id=\"fs-id1167824704572\"><span class=\"token\">\u24d2<\/span> Multiply row 3 by \\(-2\\) and add to row 1.<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167833021198\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left[\\begin{array}{cccccccccc}\\hfill 2&amp; &amp; &amp; \\hfill 1&amp; &amp; &amp; \\hfill -4&amp; &amp; &amp; \\hfill 5\\\\ \\hfill 6&amp; &amp; &amp; \\hfill -5&amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill 3\\\\ \\hfill 3&amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill 1&amp; &amp; &amp; \\hfill -1\\end{array}\\right]\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left[\\begin{array}{cccccccccc}2\\hfill &amp; &amp; &amp; \\hfill 1&amp; &amp; &amp; \\hfill -4&amp; &amp; &amp; \\hfill 5\\\\ 6\\hfill &amp; &amp; &amp; \\hfill -5&amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill 3\\\\ 3\\hfill &amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill 1&amp; &amp; &amp; \\hfill -1\\end{array}\\right]\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\left[\\begin{array}{cccccccccc}\\hfill 2&amp; &amp; &amp; \\hfill 1&amp; &amp; &amp; \\hfill -4&amp; &amp; &amp; \\hfill 5\\\\ \\hfill 6&amp; &amp; &amp; \\hfill -5&amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill 3\\\\ \\hfill -4&amp; &amp; &amp; \\hfill 7&amp; &amp; &amp; \\hfill -6&amp; &amp; &amp; \\hfill 7\\end{array}\\right]\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167824578322\"><div data-type=\"problem\" id=\"fs-id1167836626591\"><p id=\"fs-id1167836570727\">Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix: \\(\\left[\\begin{array}{cccc}\\hfill 1&amp; &amp; &amp; \\hfill 2\\\\ \\hfill -3&amp; &amp; &amp; \\hfill -4\\end{array}\\phantom{\\rule{0.5em}{0ex}}|\\phantom{\\rule{0.5em}{0ex}}\\begin{array}{c}\\hfill 5\\\\ \\hfill -1\\end{array}\\right].\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167830123187\"><p id=\"fs-id1167824649012\">Perform the needed row operations that will get the first entry in both row 2 and row 3 to be zero in the augmented matrix: \\(\\left[\\begin{array}{ccccccc}1\\hfill &amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill 3\\\\ 3\\hfill &amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill -2\\\\ 2\\hfill &amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill -4\\end{array}\\phantom{\\rule{0.5em}{0ex}}|\\phantom{\\rule{0.5em}{0ex}}\\begin{array}{c}\\hfill -4\\\\ \\hfill 5\\\\ \\hfill -1\\end{array}\\right].\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836693318\"><p id=\"fs-id1167832977161\">\\(\\left[\\begin{array}{cccccccccc}1\\hfill &amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill 3&amp; &amp; &amp; \\hfill -4\\\\ 0\\hfill &amp; &amp; &amp; \\hfill 5&amp; &amp; &amp; \\hfill -11&amp; &amp; &amp; \\hfill 17\\\\ 0\\hfill &amp; &amp; &amp; \\hfill 1&amp; &amp; &amp; \\hfill -10&amp; &amp; &amp; \\hfill 7\\end{array}\\right]\\)<\/p><\/div><\/div><p id=\"fs-id1167832982042\"><strong data-effect=\"bold\">Solve Systems of Equations Using Matrices<\/strong><\/p><p id=\"fs-id1167836601156\">In the following exercises, solve each system of equations using a matrix.<\/p><div data-type=\"exercise\" id=\"fs-id1167836573324\"><div data-type=\"problem\" id=\"fs-id1167836684975\"><p id=\"fs-id1167836440482\">\\(\\left\\{\\begin{array}{c}2x+y=2\\hfill \\\\ x-y=-2\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833007373\"><div data-type=\"problem\" id=\"fs-id1167829683972\"><p id=\"fs-id1167836498172\">\\(\\left\\{\\begin{array}{c}3x+y=2\\hfill \\\\ x-y=2\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836791945\"><p id=\"fs-id1167833019264\">\\(\\left(1,-1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826170390\"><div data-type=\"problem\" id=\"fs-id1167836476890\"><p id=\"fs-id1167836476893\">\\(\\left\\{\\begin{array}{c}\\hfill \\text{\u2212}x+2y=-2\\\\ x+y=-4\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833135703\"><div data-type=\"problem\" id=\"fs-id1167829689486\"><p id=\"fs-id1167829712505\">\\(\\left\\{\\begin{array}{c}\\hfill -2x+3y=3\\\\ x+3y=12\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836486319\"><p id=\"fs-id1167836704893\">\\(\\left(3,3\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1167833139215\">In the following exercises, solve each system of equations using a matrix.<\/p><div data-type=\"exercise\" id=\"fs-id1167836609807\"><div data-type=\"problem\" id=\"fs-id1167836609809\"><p id=\"fs-id1167836611716\">\\(\\left\\{\\begin{array}{c}2x-3y+z=19\\hfill \\\\ \\hfill -3x+y-2z=-15\\\\ x+y+z=0\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836576451\"><div data-type=\"problem\" id=\"fs-id1167836539251\"><p id=\"fs-id1167836293793\">\\(\\left\\{\\begin{array}{c}2x-y+3z=-3\\hfill \\\\ \\hfill \\text{\u2212}x+2y-z=10\\\\ x+y+z=5\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836629902\"><p id=\"fs-id1167836629904\">\\(\\left(-2,5,2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836635577\"><div data-type=\"problem\" id=\"fs-id1167836635580\"><p id=\"fs-id1167836627750\">\\(\\left\\{\\begin{array}{c}2x-6y+z=3\\hfill \\\\ 3x+2y-3z=2\\hfill \\\\ 2x+3y-2z=3\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836614234\"><div data-type=\"problem\" id=\"fs-id1167836614236\"><p id=\"fs-id1167836444948\">\\(\\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-id1167836573792\"><p id=\"fs-id1167836573794\">\\(\\left(-3,-5,4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829850155\"><div data-type=\"problem\" id=\"fs-id1167829850157\"><p id=\"fs-id1167833047400\">\\(\\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-id1167836614876\"><div data-type=\"problem\" id=\"fs-id1167836323333\"><p id=\"fs-id1167836323335\">\\(\\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-id1167829650628\"><p id=\"fs-id1167829650630\">\\(\\left(-3,2,3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829756030\"><div data-type=\"problem\" id=\"fs-id1167829756032\"><p id=\"fs-id1167836376309\">\\(\\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-id1167833059137\"><div data-type=\"problem\" id=\"fs-id1167829694210\"><p id=\"fs-id1167829694212\">\\(\\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-id1167829686580\"><p id=\"fs-id1167829686583\">\\(\\left(-2,0,-3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833047842\"><div data-type=\"problem\" id=\"fs-id1167833047844\"><p id=\"fs-id1167832945824\">\\(\\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-id1167836620917\"><div data-type=\"problem\" id=\"fs-id1167836299361\"><p id=\"fs-id1167836299363\">\\(\\left\\{\\begin{array}{c}x+2y+6z=5\\hfill \\\\ \\hfill \\text{\u2212}x+y-2z=3\\\\ x-4y-2z=1\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167836613179\">no solution<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836547142\"><div data-type=\"problem\" id=\"fs-id1167836547144\"><p id=\"fs-id1167836528457\">\\(\\left\\{\\begin{array}{c}x+2y-3z=-1\\hfill \\\\ x-3y+z=1\\hfill \\\\ 2x-y-2z=2\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836510303\"><div data-type=\"problem\" id=\"fs-id1167836689350\"><p id=\"fs-id1167836689352\">\\(\\left\\{\\begin{array}{c}4x-3y+2z=0\\hfill \\\\ \\hfill -2x+3y-7z=1\\\\ 2x-2y+3z=6\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836571645\"><p id=\"fs-id1167836433934\">no solution<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836480343\"><div data-type=\"problem\" id=\"fs-id1167836480345\"><p id=\"fs-id1167833380056\">\\(\\left\\{\\begin{array}{c}x-y+2z=-4\\hfill \\\\ 2x+y+3z=2\\hfill \\\\ \\hfill -3x+3y-6z=12\\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836531630\"><div data-type=\"problem\" id=\"fs-id1167836531632\"><p id=\"fs-id1167833056352\">\\(\\left\\{\\begin{array}{c}\\hfill \\text{\u2212}x-3y+2z=14\\\\ \\hfill \\text{\u2212}x+2y-3z=-4\\\\ 3x+y-2z=6\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836683544\"><p id=\"fs-id1167836683546\">infinitely many solutions \\(\\left(x,y,z\\right)\\) where \\(x=\\frac{1}{2}z+4;y=\\frac{1}{2}z-6;z\\) is any real number<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833022411\"><div data-type=\"problem\" id=\"fs-id1167829719183\"><p id=\"fs-id1167829719185\">\\(\\left\\{\\begin{array}{c}x+y-3z=-1\\hfill \\\\ y-z=0\\hfill \\\\ \\hfill \\text{\u2212}x+2y=1\\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167836516759\">\\(\\left\\{\\begin{array}{c}x+2y+z=4\\hfill \\\\ x+y-2z=3\\hfill \\\\ \\hfill -2x-3y+z=-7\\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836515229\"><p id=\"fs-id1167829833990\">infinitely many solutions \\(\\left(x,y,z\\right)\\) where \\(x=5z+2;y=-3z+1;z\\) is any real number<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167833047439\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167833142303\"><div data-type=\"problem\" id=\"fs-id1167833142305\"><p id=\"fs-id1167836334886\">Solve the system of equations \\(\\left\\{\\begin{array}{c}x+y=10\\hfill \\\\ x-y=6\\hfill \\end{array}\\) <span class=\"token\">\u24d0<\/span> by graphing and <span class=\"token\">\u24d1<\/span> by substitution. <span class=\"token\">\u24d2<\/span> Which method do you prefer? Why?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829651171\"><div data-type=\"problem\" id=\"fs-id1167829651173\"><p id=\"fs-id1167829716737\">Solve the system of equations \\(\\left\\{\\begin{array}{c}3x+y=12\\hfill \\\\ x=y-8\\hfill \\end{array}\\) by substitution and explain all your steps in words.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826206613\"><p id=\"fs-id1167826206615\">Answers will vary.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836319947\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167833129255\"><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-id1167832994494\" data-alt=\"This table has 4 columns 5 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 column has the following statements: Write the augmented matrix for a system of equations, Use row operations on a matrix, Solve systems of equations using matrices, Write the augmented matrix for a system of equations, Use row operations on a matrix. The remaining columns are blank.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns 5 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 column has the following statements: Write the augmented matrix for a system of equations, Use row operations on a matrix, Solve systems of equations using matrices, Write the augmented matrix for a system of equations, Use row operations on a matrix. The remaining columns are blank.\"><\/span><p><span class=\"token\">\u24d1<\/span> After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1167832999650\"><dt>matrix<\/dt><dd id=\"fs-id1167836282956\">A matrix is a rectangular array of numbers arranged in rows and columns.<\/dd><\/dl><dl id=\"fs-id1167836289514\"><dt>row-echelon form<\/dt><dd id=\"fs-id1167836510587\">A matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.<\/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>Write the augmented matrix for a system of equations<\/li>\n<li>Use row operations on a matrix<\/li>\n<li>Solve systems of equations using matrices<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836546274\" class=\"be-prepared\">\n<p id=\"fs-id1167836578744\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167836704694\" type=\"1\">\n<li>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc6c360ae83d0c568c63b55933a5f8c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#52;&#61;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"230\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167836432956\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-361b15003ae429504283783247e2921f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#50;&#53;&#112;&#43;&#48;&#46;&#50;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#46;&#50;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"211\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167836399284\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Evaluate when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01f282abd343bbe6b83c45e54b86c6ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-039e6c777b91438863be52ed95f53b5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#58;&#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;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#121;&#43;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"170\" style=\"vertical-align: -4px;\" \/>\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<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836398812\">\n<h3 data-type=\"title\">Write the Augmented Matrix for a System of Equations<\/h3>\n<p id=\"fs-id1167836731941\">Solving a system of equations can be a tedious operation where a simple mistake can wreak havoc on finding the solution. An alternative method which uses the basic procedures of elimination but with notation that is simpler is available. The method involves using a <span data-type=\"term\">matrix<\/span>. A matrix is a rectangular array of numbers arranged in rows and columns.<\/p>\n<div data-type=\"note\" id=\"fs-id1167826205478\">\n<div data-type=\"title\">Matrix<\/div>\n<p id=\"fs-id1167836508293\">A <strong data-effect=\"bold\">matrix<\/strong> is a rectangular array of numbers arranged in rows and columns.<\/p>\n<p id=\"fs-id1167833048299\">A matrix with <em data-effect=\"italics\">m<\/em> rows and <em data-effect=\"italics\">n<\/em> columns has order <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80b3e8556c79f484efa0808ac4d34842_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#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;&times;&#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;&#110;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"38\" style=\"vertical-align: 0px;\" \/> The matrix on the left below has 2 rows and 3 columns and so it has order <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-602fa60339468b2b97c575689725f9b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#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;&times;&#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;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"29\" style=\"vertical-align: 0px;\" \/> We say it is a 2 by 3 matrix.<\/p>\n<p><span data-type=\"media\" data-alt=\"Figure shows two matrices. The one on the left has the numbers minus 3, minus 2 and 2 in the first row and the numbers minus 1, 4 and 5 in the second row. The rows and columns are enclosed within brackets. Thus, it has 2 rows and 3 columns. It is labeled 2 cross 3 or 2 by 3 matrix. The matrix on the right is similar but with 3 rows and 4 columns. It is labeled 3 by 4 matrix.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows two matrices. The one on the left has the numbers minus 3, minus 2 and 2 in the first row and the numbers minus 1, 4 and 5 in the second row. The rows and columns are enclosed within brackets. Thus, it has 2 rows and 3 columns. It is labeled 2 cross 3 or 2 by 3 matrix. The matrix on the right is similar but with 3 rows and 4 columns. It is labeled 3 by 4 matrix.\" \/><\/span><\/p>\n<p id=\"fs-id1167836693648\">Each number in the matrix is called an element or entry in the matrix.<\/p>\n<\/div>\n<p id=\"fs-id1167836508919\">We will use a matrix to represent a system of linear equations. We write each equation in standard form and the coefficients of the variables and the constant of each equation becomes a row in the matrix. Each column then would be the coefficients of one of the variables in the system or the constants. A vertical line replaces the equal signs. We call the resulting matrix the augmented matrix for the system of equations.<\/p>\n<p><span data-type=\"media\" data-alt=\"The equations are 3x plus y equals minus 3 and 2x plus 3y equals 6. A 2 by 3 matrix is shown. The first row is 3, 1, minus 3. The second row is 2, 3, 6. The first column is labeled coefficients of x. The second column is labeled coefficients of y and the third is labeled constants.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_002_img_Errata.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are 3x plus y equals minus 3 and 2x plus 3y equals 6. A 2 by 3 matrix is shown. The first row is 3, 1, minus 3. The second row is 2, 3, 6. The first column is labeled coefficients of x. The second column is labeled coefficients of y and the third is labeled constants.\" \/><\/span><\/p>\n<p id=\"fs-id1167829807716\">Notice the first column is made up of all the coefficients of <em data-effect=\"italics\">x<\/em>, the second column is the all the coefficients of <em data-effect=\"italics\">y<\/em>, and the third column is all the constants.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836598326\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829685776\">\n<div data-type=\"problem\" id=\"fs-id1167836510894\">\n<p id=\"fs-id1167836573732\">Write each system of linear equations as an augmented matrix:<\/p>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6aac6809a101b92de0b50380bacb395d_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;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#50;&#120;&#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=\"43\" width=\"124\" style=\"vertical-align: -17px;\" \/><span class=\"token\">\u24d1<\/span><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-id1167836610644\">\n<p id=\"fs-id1167836486232\"><span class=\"token\">\u24d0<\/span> The second equation is not in standard form. We rewrite the second equation in standard form.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167833056775\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c1c579e4d44c107e8e932ad53057cfb_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;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#120;&#43;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#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=\"38\" width=\"158\" style=\"vertical-align: -15px;\" \/><\/div>\n<p id=\"fs-id1167829853949\">We replace the second equation with its standard form. In the augmented matrix, the first equation gives us the first row and the second equation gives us the second row. The vertical line replaces the equal signs.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836664093\" data-alt=\"The equations are 3x plus y equals minus 3 and 2x plus 3y equals 6. A 2 by 3 matrix is shown. The first row is 3, 1, minus 3. The second row is 2, 3, 6. The first column is labeled coefficients of x. The second column is labeled coefficients of y and the third is labeled constants.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_003_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are 3x plus y equals minus 3 and 2x plus 3y equals 6. A 2 by 3 matrix is shown. The first row is 3, 1, minus 3. The second row is 2, 3, 6. The first column is labeled coefficients of x. The second column is labeled coefficients of y and the third is labeled constants.\" \/><\/span><\/p>\n<p id=\"fs-id1167825836573\"><span class=\"token\">\u24d1<\/span> All three equations are in standard form. In the augmented matrix the first equation gives us the first row, the second equation gives us the second row, and the third equation gives us the third row. The vertical line replaces the equal signs.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167822916222\" data-alt=\"The equations are 6x minus 5y plus 2z equals 3, 2x plus y minus 4z equals 5 and 3x minus 3y plus z equals minus 1. A 4 by 3 matrix is shown whose first row is 6, minus 5, 2, 3. Its second row is 2, 1, minus 4, 5. Its third row is 3, minus 3, 1 and minus 1. Its first three columns are labeled x, y and z respectively.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are 6x minus 5y plus 2z equals 3, 2x plus y minus 4z equals 5 and 3x minus 3y plus z equals minus 1. A 4 by 3 matrix is shown whose first row is 6, minus 5, 2, 3. Its second row is 2, 1, minus 4, 5. Its third row is 3, minus 3, 1 and minus 1. Its first three columns are labeled x, y and z respectively.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833059065\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829692719\">\n<div data-type=\"problem\" id=\"fs-id1167836774865\">\n<p id=\"fs-id1167829743626\">Write each system of linear equations as an augmented matrix:<\/p>\n<p id=\"fs-id1167836522302\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7faa6d1b829c54ab45bf182eaa4ffc3c_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;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#61;&#45;&#53;&#121;&#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=\"43\" width=\"125\" style=\"vertical-align: -17px;\" \/><span class=\"token\">\u24d1<\/span><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 data-type=\"solution\" id=\"fs-id1167836292577\">\n<p id=\"fs-id1167833310842\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d21277397dda1fd25c20e65f7f43fa50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#92;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"164\" style=\"vertical-align: -17px;\" \/><\/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-37ca3f842026a45fb837122d61abc23f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#38;&#32;&#38;&#32;&#38;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#56;&#92;&#92;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#55;&#92;&#92;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"239\" style=\"vertical-align: -28px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829829156\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833366592\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836546221\">Write each system of linear equations as an augmented matrix:<\/p>\n<p id=\"fs-id1167829753078\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d413db6e5b69d909477906cec9e99b25_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;&#61;&#45;&#57;&#121;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#55;&#120;&#43;&#53;&#121;&#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=\"43\" width=\"133\" style=\"vertical-align: -17px;\" \/><span class=\"token\">\u24d1<\/span><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-id1167836527130\">\n<p id=\"fs-id1167829598310\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34e8e262641468a2d024cde1dc3ffdce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#49;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#92;&#92;&#32;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"173\" style=\"vertical-align: -17px;\" \/><\/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-bae0ea0068aaadb8a2874b470e66c325_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#92;&#92;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#92;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#55;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"253\" style=\"vertical-align: -28px;\" \/><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833335409\">It is important as we solve systems of equations using matrices to be able to go back and forth between the system and the matrix. The next example asks us to take the information in the matrix and write the system of equations.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829908050\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829721061\">\n<div data-type=\"problem\" id=\"fs-id1167829579076\">\n<p id=\"fs-id1167829877945\">Write the system of equations that corresponds to the augmented matrix:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b3b7839f73203558f8aa8ac620a3d6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"315\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833335007\">\n<p>We remember that each row corresponds to an equation and that each entry is a coefficient of a variable or the constant. The vertical line replaces the equal sign. Since this matrix is a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f2a09c3f51be46b69a40bb259dc3e756_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#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;&times;&#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;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"25\" style=\"vertical-align: -1px;\" \/>, we know it will translate into a system of three equations with three variables.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167823183143\" data-alt=\"A 3 by 4 matrix is shown. Its first row is 4, minus 3, 3, minus 1. Its second row is 1, 2, minus 1, 2. Its third row is minus 2, minus 1, 3, minus 4. The three equations are 4x minus 3y plus 3z equals minus 1, x plus 2y minus z equals 2 and minus 2x minus y plus 3z equals minus 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A 3 by 4 matrix is shown. Its first row is 4, minus 3, 3, minus 1. Its second row is 1, 2, minus 1, 2. Its third row is minus 2, minus 1, 3, minus 4. The three equations are 4x minus 3y plus 3z equals minus 1, x plus 2y minus z equals 2 and minus 2x minus y plus 3z equals minus 4.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167824755056\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167826188018\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836743351\">Write the system of equations that corresponds to the augmented matrix: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-96d94e4f39452207b01cebda7cb0a339_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#92;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#92;&#92;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"251\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167833350368\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bd6e3a812962868405257dc1ff68b19_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;&#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;&#120;&#45;&#121;&#43;&#50;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#121;&#45;&#50;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#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;&#52;&#120;&#45;&#121;&#43;&#50;&#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=\"150\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167824617356\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836497729\">\n<div data-type=\"problem\" id=\"fs-id1167836693299\">\n<p id=\"fs-id1167836691914\">Write the system of equations that corresponds to the augmented matrix: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16792cd6be09b002672eb5077a763ad8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#92;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#56;&#92;&#92;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"237\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829717165\">\n<p id=\"fs-id1167836293044\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b48e7bcd9eae0d34a2d3274b25f1cc3_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;&#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;&#120;&#43;&#121;&#43;&#122;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#45;&#122;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#43;&#121;&#45;&#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=\"143\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\">Use Row Operations on a Matrix<\/h3>\n<p>Once a system of equations is in its augmented matrix form, we will perform operations on the rows that will lead us to the solution.<\/p>\n<p id=\"fs-id1167829784535\">To solve by elimination, it doesn\u2019t matter which order we place the equations in the system. Similarly, in the matrix we can interchange the rows.<\/p>\n<p id=\"fs-id1167836393210\">When we solve by elimination, we often multiply one of the equations by a constant. Since each row represents an equation, and we can multiply each side of an equation by a constant, similarly we can multiply each entry in a row by any real number except 0.<\/p>\n<p>In elimination, we often add a multiple of one row to another row. In the matrix we can replace a row with its sum with a multiple of another row.<\/p>\n<p id=\"fs-id1167836538695\">These actions are called row operations and will help us use the matrix to solve a system of equations.<\/p>\n<div data-type=\"note\" id=\"fs-id1167833061563\">\n<div data-type=\"title\">Row Operations<\/div>\n<p id=\"fs-id1167836456012\">In a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix.<\/p>\n<ol id=\"fs-id1167829624876\" type=\"1\">\n<li>Interchange any two rows.<\/li>\n<li>Multiply a row by any real number except 0.<\/li>\n<li>Add a nonzero multiple of one row to another row.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167836515322\">Performing these operations is easy to do but all the arithmetic can result in a mistake. If we use a system to record the row operation in each step, it is much easier to go back and check our work.<\/p>\n<p id=\"fs-id1167824732306\">We use capital letters with subscripts to represent each row. We then show the operation to the left of the new matrix. To show interchanging a row:<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167829807711\" data-alt=\"A 2 by 3 matrix is shown. Its first row, labeled R2 is 2, minus 1, 2. Its second row, labeled R1 is 5, minus 3, minus 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_006_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A 2 by 3 matrix is shown. Its first row, labeled R2 is 2, minus 1, 2. Its second row, labeled R1 is 5, minus 3, minus 1.\" \/><\/span><\/p>\n<p id=\"fs-id1167833129901\">To multiply row 2 by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/>:<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836541327\" data-alt=\"A 2 by 3 matrix is shown. Its first row is 5, minus 3, minus 1. Its second row is 2, minus 1, 2. An arrow point from this matrix to another one on the right. The first row of the new matrix is the same. The second row is preceded by minus 3 R2. It is minus 6, 3, minus 6.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A 2 by 3 matrix is shown. Its first row is 5, minus 3, minus 1. Its second row is 2, minus 1, 2. An arrow point from this matrix to another one on the right. The first row of the new matrix is the same. The second row is preceded by minus 3 R2. It is minus 6, 3, minus 6.\" \/><\/span><\/p>\n<p id=\"fs-id1167836558970\">To multiply row 2 by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> and add it to row 1:<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167829829286\" data-alt=\"A 2 by 3 matrix is shown. Its first row is 5, minus 3, minus 1. Its second row is 2, minus 1, 2. An arrow point from this matrix to another one on the right. The first row of the new matrix is preceded by minus 3 R2 plus R1. It is minus 1, 0, minus 7. The second row is 2, minus 1, 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_008_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A 2 by 3 matrix is shown. Its first row is 5, minus 3, minus 1. Its second row is 2, minus 1, 2. An arrow point from this matrix to another one on the right. The first row of the new matrix is preceded by minus 3 R2 plus R1. It is minus 1, 0, minus 7. The second row is 2, minus 1, 2.\" \/><\/span><\/p>\n<div data-type=\"example\" id=\"fs-id1167833350263\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167824780787\">\n<div data-type=\"problem\" id=\"fs-id1167836499275\">\n<p id=\"fs-id1167826205547\">Perform the indicated operations on the augmented matrix:<\/p>\n<p id=\"fs-id1167829748933\"><span class=\"token\">\u24d0<\/span> Interchange rows 2 and 3.<\/p>\n<p id=\"fs-id1167833024294\"><span class=\"token\">\u24d1<\/span> Multiply row 2 by 5.<\/p>\n<p id=\"fs-id1167829739413\"><span class=\"token\">\u24d2<\/span> Multiply row 3 by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> and add to row 1.<\/p>\n<div data-type=\"equation\" id=\"fs-id1171792520714\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-76c0d6ba798400e2f19dd307c860509e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"290\" style=\"vertical-align: -28px;\" \/><\/div>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836645967\">\n<p id=\"fs-id1167833024483\"><span class=\"token\">\u24d0<\/span> We interchange rows 2 and 3.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167824669254\" data-alt=\"Two 3 by 4 matrices are shown. In the one on the left, the first row is 6, minus 5, 2, 3. The second row is 2, 1, minus 4, 5. The third row is 3, minus 3, 1, minus 1. The second matrix is similar except that rows 2 and 3 are interchanged.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Two 3 by 4 matrices are shown. In the one on the left, the first row is 6, minus 5, 2, 3. The second row is 2, 1, minus 4, 5. The third row is 3, minus 3, 1, minus 1. The second matrix is similar except that rows 2 and 3 are interchanged.\" \/><\/span><\/p>\n<p id=\"fs-id1167836328953\"><span class=\"token\">\u24d1<\/span> We multiply row 2 by 5.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" data-alt=\"Two 3 by 4 matrices are shown. In the one on the left, the first row is 6, minus 5, 2, 3. The second row is 2, 1, minus 4, 5. The third row is 3, minus 3, 1, minus 1. The second matrix is similar to the first except that row 2, preceded by 5 R2, is 10, 5, minus 20, 25.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_010_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Two 3 by 4 matrices are shown. In the one on the left, the first row is 6, minus 5, 2, 3. The second row is 2, 1, minus 4, 5. The third row is 3, minus 3, 1, minus 1. The second matrix is similar to the first except that row 2, preceded by 5 R2, is 10, 5, minus 20, 25.\" \/><\/span><\/p>\n<p id=\"fs-id1167829712000\"><span class=\"token\">\u24d2<\/span> We multiply row 3 by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> and add to row 1.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167836292453\" data-alt=\"In the 3 by 4 matrix, the first row is 6, minus 5, 2, 3. The second row is 2, 1, minus 4, 5. The third row is 3, minus 3, 1, minus 1. Performing the operation minus 2 R3 plus R1 on the first row, the first row becomes 6 plus minus 2 times 3, minus 5 plus minus 2 times minus 3, 2 plus minus 2 times 1 and 3 plus minus 2 times minus 1. This becomes 0, 1, 0, 5. The remaining 2 rows of the new matrix are the same.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_011_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In the 3 by 4 matrix, the first row is 6, minus 5, 2, 3. The second row is 2, 1, minus 4, 5. The third row is 3, minus 3, 1, minus 1. Performing the operation minus 2 R3 plus R1 on the first row, the first row becomes 6 plus minus 2 times 3, minus 5 plus minus 2 times minus 3, 2 plus minus 2 times 1 and 3 plus minus 2 times minus 1. This becomes 0, 1, 0, 5. The remaining 2 rows of the new matrix are the same.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829786590\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836335371\">\n<div data-type=\"problem\" id=\"fs-id1167829594338\">\n<p id=\"fs-id1167825993101\">Perform the indicated operations on the augmented matrix:<\/p>\n<p id=\"fs-id1167829688134\"><span class=\"token\">\u24d0<\/span> Interchange rows 1 and 3.<\/p>\n<p id=\"fs-id1167836334004\"><span class=\"token\">\u24d1<\/span> Multiply row 3 by 3.<\/p>\n<p id=\"fs-id1167825691607\"><span class=\"token\">\u24d2<\/span> Multiply row 3 by 2 and add to row 2.<\/p>\n<p id=\"fs-id1167833224470\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ccd5e762fe40098f402fcc8ee4b6f8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"304\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167829627951\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c20e17366ac05c42a03b955bf2b5d479_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"266\" style=\"vertical-align: -28px;\" \/><\/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-092076634f77e6cf1c1066c99b9150c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#92;&#32;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"266\" style=\"vertical-align: -28px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5db3467290bef2821fa3e99f76e0c6fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#54;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#56;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#53;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"355\" style=\"vertical-align: -28px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826172223\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836366482\">\n<div data-type=\"problem\" id=\"fs-id1167836732417\">\n<p id=\"fs-id1167836477875\">Perform the indicated operations on the augmented matrix:<\/p>\n<p id=\"fs-id1167836300769\"><span class=\"token\">\u24d0<\/span> Interchange rows 1 and 2,<\/p>\n<p id=\"fs-id1167836611076\"><span class=\"token\">\u24d1<\/span> Multiply row 1 by 2,<\/p>\n<p id=\"fs-id1167829859334\"><span class=\"token\">\u24d2<\/span> Multiply row 2 by 3 and add to row 1.<\/p>\n<p id=\"fs-id1167836352756\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4ab99bf665ff3c633733972189735ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"290\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829841016\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1b227e2683cb7f5798bd5cbcd11d91f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#92;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#92;&#32;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"253\" style=\"vertical-align: -28px;\" \/><\/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-426c9eeb4d9e4e11ff7f5313e4b06335_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#92;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#92;&#32;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"253\" style=\"vertical-align: -28px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9af45aefa9dec11e667b5e179c6a9a8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#49;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#55;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#56;&#92;&#92;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#92;&#32;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"270\" style=\"vertical-align: -28px;\" \/><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836515290\">Now that we have practiced the row operations, we will look at an augmented matrix and figure out what operation we will use to reach a goal. This is exactly what we did when we did elimination. We decided what number to multiply a row by in order that a variable would be eliminated when we added the rows together.<\/p>\n<p id=\"fs-id1167833377541\">Given this system, what would you do to eliminate <em data-effect=\"italics\">x<\/em>?<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167829786049\" data-alt=\"The two equations are x minus y equals 2 and 4x minus 8y equals 0. Multiplying the first by minus 4, we get minus 4x plus 4y equals minus 8. Adding this to the second equation we get minus 4y equals minus 8.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_012_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The two equations are x minus y equals 2 and 4x minus 8y equals 0. Multiplying the first by minus 4, we get minus 4x plus 4y equals minus 8. Adding this to the second equation we get minus 4y equals minus 8.\" \/><\/span><\/p>\n<p id=\"fs-id1167836540507\">This next example essentially does the same thing, but to the matrix.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836409774\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836319795\">\n<p id=\"fs-id1167836768391\">Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8c4780f226b85b7fc15619da6b07ebef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#56;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"213\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167829624231\">To make the 4 a 0, we could multiply row 1 by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/> and then add it to row 2.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167833365764\" data-alt=\"The 2 by 3 matrix is 1, minus 1, 2 and 4, minus 8, 0. Performing the operation minus 4R1 plus R2 on row 2, the second row of the new matrix becomes 0, minus 4, minus 8. The first row remains the same.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_013_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The 2 by 3 matrix is 1, minus 1, 2 and 4, minus 8, 0. Performing the operation minus 4R1 plus R2 on row 2, the second row of the new matrix becomes 0, minus 4, minus 8. The first row remains the same.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836524340\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836630643\">\n<div data-type=\"problem\" id=\"fs-id1167833279820\">\n<p id=\"fs-id1167833196797\">Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3325369d3a1e67d1f9f8c5cd3a0f58db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"213\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833310074\">\n<p id=\"fs-id1167833025573\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-30f4afbf06773cad2cb84f669c32e8b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#92;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"178\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836531933\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829831866\">\n<div data-type=\"problem\" id=\"fs-id1167833202534\">\n<p id=\"fs-id1167829586303\">Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0251e91da2e4cda4941d22d7eb6dd85_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"226\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836535460\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3fd14b3c0e90491c0c99ec988e79ab22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#92;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#56;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"164\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\">Solve Systems of Equations Using Matrices<\/h3>\n<p id=\"fs-id1167833338960\">To solve a system of equations using matrices, we transform the augmented matrix into a matrix in <span data-type=\"term\">row-echelon form<\/span> using row operations. For a consistent and independent system of equations, its <span data-type=\"term\" class=\"no-emphasis\">augmented matrix<\/span> is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836363886\">\n<div data-type=\"title\">Row-Echelon Form<\/div>\n<p id=\"fs-id1167829578768\">For a consistent and independent system of equations, its augmented matrix is in <strong data-effect=\"bold\">row-echelon form<\/strong> when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.<\/p>\n<p><span data-type=\"media\" data-alt=\"A 2 by 3 matrix is shown on the left. Its first row is 1, a, b. Its second row is 0, 1, c. An arrow points diagonally down and right, overlapping both the 1s in the matrix. A 3 by 4 matrix is shown on the right. Its first row is 1, a, b, d. Its second row is 0, 1, c, e. Its third row is 0, 0, 1, f. An arrow points diagonally down and right, overlapping all the 1s in the matrix. a, b, c, d, e, f are real numbers.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_014_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A 2 by 3 matrix is shown on the left. Its first row is 1, a, b. Its second row is 0, 1, c. An arrow points diagonally down and right, overlapping both the 1s in the matrix. A 3 by 4 matrix is shown on the right. Its first row is 1, a, b, d. Its second row is 0, 1, c, e. Its third row is 0, 0, 1, f. An arrow points diagonally down and right, overlapping all the 1s in the matrix. a, b, c, d, e, f are real numbers.\" \/><\/span><\/div>\n<p>Once we get the augmented matrix into row-echelon form, we can write the equivalent system of equations and read the value of at least one variable. We then substitute this value in another equation to continue to solve for the other variables. This process is illustrated in the next example.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829692902\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Solve a System of Equations Using a Matrix<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829743345\">\n<div data-type=\"problem\" id=\"fs-id1167822916232\">\n<p id=\"fs-id1167830093682\">Solve the system of equations using a matrix: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8191171d32ab5696dd8f1f670cfb3148_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;&#52;&#121;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#50;&#121;&#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=\"43\" width=\"123\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167822996975\"><span data-type=\"media\" id=\"fs-id1167836525799\" data-alt=\"The equations are 3x plus 4y equals 5 and x plus 2y equals 1. Step 1. Write the augmented matrix for the system of equations. We get a 2 by 3 matrix with first row 3, 4, 5 and second row 1, 2, 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_015a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are 3x plus 4y equals 5 and x plus 2y equals 1. Step 1. Write the augmented matrix for the system of equations. We get a 2 by 3 matrix with first row 3, 4, 5 and second row 1, 2, 1.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167829828529\" data-alt=\"Step 2. Using row operations get the entry in row 1, column 1 to be 1. Interchange rows R1 and R2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_015b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2. Using row operations get the entry in row 1, column 1 to be 1. Interchange rows R1 and R2.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167833382010\" data-alt=\"Step 3. Using row operations, get zeros in column 1 below the 1. Multiply row 1 by minus 3 and add it to row 2. Row 2 becomes 0, minus 2, 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_015c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3. Using row operations, get zeros in column 1 below the 1. Multiply row 1 by minus 3 and add it to row 2. Row 2 becomes 0, minus 2, 2.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836692585\" data-alt=\"Step 4. Using row operations, get the entry in row 2, column 2 to be 1. Multiply row 2 by minus half. Row 2 becomes 0, 1, minus 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_015d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4. Using row operations, get the entry in row 2, column 2 to be 1. Multiply row 2 by minus half. Row 2 becomes 0, 1, minus 1.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167833021336\" data-alt=\"Step 5. Continue the process until the matrix is in row-echelon form. The matrix is now in row-echelon form.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_015e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5. Continue the process until the matrix is in row-echelon form. The matrix is now in row-echelon form.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836713900\" data-alt=\"Step 6. Write the corresponding system of equations. We get x plus 2y equals 1 and y equals minus 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_015f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 6. Write the corresponding system of equations. We get x plus 2y equals 1 and y equals minus 1.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836688662\" data-alt=\"Step 7. Use substitution to find the remaining variables. Substitute y equals negative 1 into x plus 2y equals 1. X plus 2 times negative 1 equals 1. X minus 2 equals 1. We get x equal to 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_015g_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 7. Use substitution to find the remaining variables. Substitute y equals negative 1 into x plus 2y equals 1. X plus 2 times negative 1 equals 1. X minus 2 equals 1. We get x equal to 3.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167832930182\" data-alt=\"Step 8. Write the solution as an ordered pair or triple. Ordered pair is (3, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_015h_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 8. Write the solution as an ordered pair or triple. Ordered pair is (3, negative 1).\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836542466\" data-alt=\"Step 9. Check that the solution makes the original equations true.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_015i_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 9. Check that the solution makes the original equations true.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836484478\">Solve the system of equations using a matrix: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93c77aec095fb80605576f00f515b39d_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;&#121;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#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=\"43\" width=\"114\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836533019\">\n<p id=\"fs-id1167825823995\">The solution is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65fe2760e5f7600eb05caea522df3c36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836621323\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836518516\">\n<div data-type=\"problem\" id=\"fs-id1167836575707\">\n<p id=\"fs-id1167824852264\">Solve the system of equations using a matrix: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9552aaefd902bb9bb31bb6cb9e343318_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;&#121;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#121;&#61;&#45;&#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=\"43\" width=\"128\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836552453\">The solution is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-632eac8be1e0cf0c93c74b5d38f51abe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833227319\">The steps are summarized here.<\/p>\n<div data-type=\"note\" id=\"fs-id1167833023848\" class=\"howto\">\n<div data-type=\"title\">Solve a system of equations using matrices.<\/div>\n<ol id=\"fs-id1167836485160\" type=\"1\" class=\"stepwise\">\n<li>Write the augmented matrix for the system of equations.<\/li>\n<li>Using row operations get the entry in row 1, column 1 to be 1.<\/li>\n<li>Using row operations, get zeros in column 1 below the 1.<\/li>\n<li>Using row operations, get the entry in row 2, column 2 to be 1.<\/li>\n<li>Continue the process until the matrix is in row-echelon form.<\/li>\n<li>Write the corresponding system of equations.<\/li>\n<li>Use substitution to find the remaining variables.<\/li>\n<li>Write the solution as an ordered pair or triple.<\/li>\n<li>Check that the solution makes the original equations true.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167833058881\">Here is a visual to show the order for getting the 1\u2019s and 0\u2019s in the proper position for row-echelon form.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836295125\" data-alt=\"The figure shows 3 steps for a 2 by 3 matrix and 6 steps for a 3 by 4 matrix. For the former, step 1 is to get a 1 in row 1 column 1. Step to is to get a 0 is row 2 column 1. Step 3 is to get a 1 in row 2 column 2. For a 3 by 4 matrix, step 1 is to get a 1 in row 1 column 1. Step 2 is to get a 0 in row 2 column 1. Step 3 is to get a 0 in row 3 column 1. Step 4 is to get a 1 in row 2 column 2. Step 5 is to get a 0 in row 3 column 2. Step 6 is to get a 1 in row 3 column 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_016_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows 3 steps for a 2 by 3 matrix and 6 steps for a 3 by 4 matrix. For the former, step 1 is to get a 1 in row 1 column 1. Step to is to get a 0 is row 2 column 1. Step 3 is to get a 1 in row 2 column 2. For a 3 by 4 matrix, step 1 is to get a 1 in row 1 column 1. Step 2 is to get a 0 in row 2 column 1. Step 3 is to get a 0 in row 3 column 1. Step 4 is to get a 1 in row 2 column 2. Step 5 is to get a 0 in row 3 column 2. Step 6 is to get a 1 in row 3 column 3.\" \/><\/span><\/p>\n<p id=\"fs-id1168757403507\">We use the same procedure when the system of equations has three equations.<\/p>\n<div data-type=\"example\" id=\"fs-id1167832984010\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829812225\">\n<div data-type=\"problem\" id=\"fs-id1167833378931\">\n<p id=\"fs-id1167829690073\">Solve the system of equations using a matrix: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4a5e5a785a1993a5a27d4ea6b36501a_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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"178\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<table id=\"fs-id1167829714495\" class=\"unnumbered unstyled can-break\" summary=\"The equations are 3x plus 8y plus 2z equals minus 5, 2x plus 5y minus 3z equals 0, x plus 2y minus 2z equals minus 1. Write the augmented matrix for the equations. Row 1 is 3, 8, 2, minus 5. Row 2 is 2, 5, minus 3, 0. Row 3 is 1, 2, minus 2, minus 1. Interchange row 1 and 3 to get the entry in row 1, column 1 to be 1. Use operation minus 2R1 plus R2 on row 2. Use operation minus 3R1 plus R3 on row 3. Use operation minus 2R2 plus R3 on row 3. Use operation 1 upon 6 R3 on row 3. The matrix is now in row-echelon form. The corresponding system of equations is x plus 2y minus 2z equals minus 1, y plus z equals 2 and z equals minus 1. Using substitution, we get y equal to 3 and x equal to minus 9. The solution is minus 9, 3, minus 1. Check that the original equations hold true.\" 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\"><span data-type=\"media\" id=\"fs-id1167836309310\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the augmented matrix for the equations.<\/td>\n<td 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_05_017b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Interchange row 1 and 3 to get the entry in<\/p>\n<div data-type=\"newline\"><\/div>\n<p>row 1, column 1 to be 1.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836329209\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Using row operations, get zeros in column 1 below the 1.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836531979\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836340874\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The entry in row 2, column 2 is now 1.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Continue the process until the matrix<\/p>\n<div data-type=\"newline\"><\/div>\n<p>is in row-echelon form.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167823026715\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829748610\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The matrix is now in row-echelon form.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829744236\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the corresponding system of equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824852333\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017i_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 substitution to find the remaining variables.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824740544\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829624548\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><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_05_017m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the solution as an ordered pair or triple.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836477846\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_017n_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check that the solution makes the original equations true.<\/td>\n<td data-valign=\"top\" data-align=\"left\">We leave the check for you.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829807014\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836548820\">\n<div data-type=\"problem\" id=\"fs-id1167833380079\">\n<p>Solve the system of equations using a matrix: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a7ee919e7085cea77ea01af1814efdbf_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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"169\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833020740\">\n<p id=\"fs-id1167829709377\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c2a86819bbfea73aa6b5a3a7e25853b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#45;&#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<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833009146\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836685150\">\n<div data-type=\"problem\" id=\"fs-id1167836388313\">\n<p id=\"fs-id1167836330398\">Solve the system of equations using a matrix: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-75eafa26a89429b627cf680bd1ec8c71_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;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#120;&#43;&#121;&#43;&#122;&#61;&#45;&#52;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#50;&#121;&#45;&#50;&#122;&#61;&#49;&#92;&#92;&#32;&#50;&#120;&#45;&#121;&#45;&#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=\"174\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836299894\">\n<p id=\"fs-id1167833024271\"><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;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833158290\">So far our work with matrices has only been with systems that are consistent and independent, which means they have exactly one solution. Let\u2019s now look at what happens when we use a matrix for a dependent or inconsistent system.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836693828\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836357145\">\n<div data-type=\"problem\" id=\"fs-id1167836697732\">\n<p id=\"fs-id1167829688774\">Solve the system of equations using a matrix: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eeeba32650f257a826d811edeab46850_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;&#43;&#51;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#51;&#121;&#43;&#53;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#52;&#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=\"155\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826172431\">\n<table id=\"fs-id1167836320619\" class=\"unnumbered unstyled can-break\" summary=\"The equations are x plus y plus 3z equals 0, x plus 3y plus 5z equals 0 and 2x plus 4z equals 1. The first row of the augmented matrix is 1, 1, 3, 0. Row 2 is 1, 3, 5, 0. Row 3 is 2, 0, 4, 1. Use row operation minus 1R1 plus R2 on row 2. Use operation minus 2R1 plus R3 on row 3. Use operation half R2 on row 2. Use operation 2R2 plus R3. The corresponding equations are x plus y plus 3z equals 0, y plus z equals 0 and 0 not equal to 1.\" 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\"><span data-type=\"media\" id=\"fs-id1167829693728\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_018a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the augmented matrix for the equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836552968\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_018b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The entry in row 1, column 1 is 1.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Using row operations, get zeros in column 1 below the 1.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836697832\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_018c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836628663\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_018d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Continue the process until the matrix is in row-echelon form.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836339886\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_018e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply row 2 by 2 and add it to row 3.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836605395\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_018f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">At this point, we have all zeros on the left of row 3.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the corresponding system of equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829597744\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_018g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2e1a552e64cc3c55f0f6ffb31982ecd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#110;&#101;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"41\" style=\"vertical-align: -4px;\" \/> we have a false statement. Just as when we solved a system using other methods, this tells us we have an inconsistent system. There is no solution.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836507507\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836656607\">Solve the system of equations using a matrix: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-184fcaba83dc7b9e29bbe18dc0619828_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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#120;&#43;&#121;&#45;&#122;&#61;&#50;&#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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"160\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836697856\">\n<p id=\"fs-id1167836556734\">no solution<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836597057\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836699123\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836311815\">Solve the system of equations using a matrix: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e3c86ad86e9d71df94248bf5f720230_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;&#52;&#121;&#45;&#51;&#122;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#45;&#122;&#61;&#45;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#121;&#45;&#50;&#122;&#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=\"178\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836585335\">\n<p id=\"fs-id1167829686514\">no solution<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833019836\">The last system was inconsistent and so had no solutions. The next example is dependent and has infinitely many solutions.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836555721\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167832925613\">\n<div data-type=\"problem\" id=\"fs-id1167836535076\">\n<p id=\"fs-id1167829807672\">Solve the system of equations using a matrix: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d4f9bbfebd9794f88db4266b5953c14_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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"164\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836484491\">\n<table id=\"fs-id1167836701231\" class=\"unnumbered unstyled can-break\" summary=\"The equations are x minus 2y plus 3z equals 1, x plus y minus 3z equals 7 and 3x minus 4y plus 5z equals 7. The augmented matrix is: row 1: 1, minus 2, 3, 1, row 2: 1, 1, minus 3, 7, row 3: 3, minus 4, 5, 7. Use operation minus 1R1 plus R2 on row 2. use operation minus 3 R1 plus R3 on row 3. Use operation 1 upon 3 R2 on row 2. Use operation minus 2R2 plus R3 on row 3. The corresponding equations are x minus 2y plus 3z is 1, y minus 2z is 2 and 0 is 0. Since 0 is 0 we have a true statement. Just as when we solved by substitution, this tells us we have a dependent system. There are infinitely many solutions. Solving for y in second equation, we get y equal to 2z plus 2. Substituting this in the first equation, we get x equal to z plus 5. The system has infinitely many solutions x, y, z where x is z plus 5 and y is 2z plus 2 and z is any real number.\" 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\"><span data-type=\"media\" id=\"fs-id1167836662838\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the augmented matrix for the equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836352824\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The entry in row 1, column 1 is 1.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Using row operations, get zeros in column 1 below the 1.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836529008\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td 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_05_019d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Continue the process until the matrix is in row-echelon form.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833256119\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply row 2 by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> and add it to row 3.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829741959\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">At this point, we have all zeros in the bottom row.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the corresponding system of equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833245758\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Since <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;\" \/> we have a true statement. Just as when we solved by substitution, this tells us we have a dependent system. There are infinitely many solutions.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">y<\/em> in terms of <em data-effect=\"italics\">z<\/em> in the second equation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836493258\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019h_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 the first equation for <em data-effect=\"italics\">x<\/em> in terms of <em data-effect=\"italics\">z<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836707050\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/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-eb9414307ef77975c75e3b1fe7c9d75d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#122;&#43;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"86\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829720122\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829721214\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836649335\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829720944\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_019m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">The system has infinitely many solutions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-76fd1815cc64c58044691df3370bcb30_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;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" style=\"vertical-align: -4px;\" \/> where<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8846e31b3673cf83e9e61d021095a711_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#122;&#43;&#53;&#59;&#121;&#61;&#50;&#122;&#43;&#50;&#59;&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"179\" style=\"vertical-align: -4px;\" \/> is any real number.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833207815\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829692179\">\n<div data-type=\"problem\" id=\"fs-id1167836600243\">\n<p id=\"fs-id1167829745733\">Solve the system of equations using a matrix: <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-id1167836293696\">\n<p>infinitely many solutions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-76fd1815cc64c58044691df3370bcb30_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;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" 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\" id=\"fs-id1167836315181\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836349086\">\n<div data-type=\"problem\" id=\"fs-id1167836310628\">\n<p id=\"fs-id1167833158281\">Solve the system of equations using a matrix: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e806e800188261ee6de5e7b6717eede_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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#50;&#121;&#45;&#51;&#122;&#61;&#45;&#52;&#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=\"169\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829843556\">\n<p id=\"fs-id1167833059149\">infinitely many solutions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-76fd1815cc64c58044691df3370bcb30_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;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" 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 data-type=\"note\" id=\"fs-id1167829748849\" class=\"media-2\">\n<p id=\"fs-id1167833274200\">Access this online resource for additional instruction and practice with Gaussian Elimination.<\/p>\n<ul id=\"fs-id1169145728353\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37GaussElim\">Gaussian Elimination<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836626308\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167836293453\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Matrix:<\/strong> A matrix is a rectangular array of numbers arranged in rows and columns. A matrix with <em data-effect=\"italics\">m<\/em> rows and <em data-effect=\"italics\">n<\/em> columns has <em data-effect=\"italics\">order<\/em> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80b3e8556c79f484efa0808ac4d34842_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#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;&times;&#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;&#110;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"38\" style=\"vertical-align: 0px;\" \/> The matrix on the left below has 2 rows and 3 columns and so it has order <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-602fa60339468b2b97c575689725f9b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#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;&times;&#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;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"29\" style=\"vertical-align: 0px;\" \/> We say it is a 2 by 3 matrix.\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167833385766\" data-alt=\"Figure shows two matrices. The one on the left has the numbers minus 3, minus 2 and 2 in the first row and the numbers minus 1, 4 and 5 in the second row. The rows and columns are enclosed within brackets. Thus, it has 2 rows and 3 columns. It is labeled 2 cross 3 or 2 by 3 matrix. The matrix on the right is similar but with 3 rows and 4 columns. It is labeled 3 by 4 matrix.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_020_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows two matrices. The one on the left has the numbers minus 3, minus 2 and 2 in the first row and the numbers minus 1, 4 and 5 in the second row. The rows and columns are enclosed within brackets. Thus, it has 2 rows and 3 columns. It is labeled 2 cross 3 or 2 by 3 matrix. The matrix on the right is similar but with 3 rows and 4 columns. It is labeled 3 by 4 matrix.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p> Each number in the matrix is called an <em data-effect=\"italics\">element<\/em> or <em data-effect=\"italics\">entry<\/em> in the matrix.<\/li>\n<li><strong data-effect=\"bold\">Row Operations:<\/strong> In a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix.\n<ul id=\"fs-id1167836613366\" data-bullet-style=\"bullet\">\n<li>Interchange any two rows<\/li>\n<li>Multiply a row by any real number except 0<\/li>\n<li>Add a nonzero multiple of one row to another row<\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">Row-Echelon Form:<\/strong> For a consistent and independent system of equations, its augmented matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167833316687\" data-alt=\"Figure shows two matrices. The one on the left has the numbers minus 3, minus 2 and 2 in the first row and the numbers minus 1, 4 and 5 in the second row. The rows and columns are enclosed within brackets. Thus, it has 2 rows and 3 columns. It is labeled 2 cross 3 or 2 by 3 matrix. The matrix on the right is similar but with 3 rows and 4 columns. It is labeled 3 by 4 matrix.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_05_021_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows two matrices. The one on the left has the numbers minus 3, minus 2 and 2 in the first row and the numbers minus 1, 4 and 5 in the second row. The rows and columns are enclosed within brackets. Thus, it has 2 rows and 3 columns. It is labeled 2 cross 3 or 2 by 3 matrix. The matrix on the right is similar but with 3 rows and 4 columns. It is labeled 3 by 4 matrix.\" \/><\/span> <\/li>\n<li><strong data-effect=\"bold\">How to solve a system of equations using matrices.<\/strong>\n<ol id=\"fs-id1167836386945\" type=\"1\" class=\"stepwise\">\n<li>Write the augmented matrix for the system of equations.<\/li>\n<li>Using row operations get the entry in row 1, column 1 to be 1.<\/li>\n<li>Using row operations, get zeros in column 1 below the 1.<\/li>\n<li>Using row operations, get the entry in row 2, column 2 to be 1.<\/li>\n<li>Continue the process until the matrix is in row-echelon form.<\/li>\n<li>Write the corresponding system of equations.<\/li>\n<li>Use substitution to find the remaining variables.<\/li>\n<li>Write the solution as an ordered pair or triple.<\/li>\n<li>Check that the solution makes the original equations true.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167833025558\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167836449701\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167836547542\"><strong data-effect=\"bold\">Write the Augmented Matrix for a System of Equations<\/strong><\/p>\n<p id=\"fs-id1167826171377\">In the following exercises, write each system of linear equations as an augmented matrix.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836523331\">\n<div data-type=\"problem\" id=\"fs-id1167829844031\">\n<p id=\"fs-id1167836620562\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d82b451ef2a677545922f3074e891b7_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;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#121;&#61;&#50;&#120;&#43;&#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=\"43\" width=\"115\" style=\"vertical-align: -17px;\" \/><\/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-18645470d80dab2f4e574fa4faf71b11_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;&#51;&#121;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#121;&#45;&#51;&#122;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#121;&#43;&#50;&#122;&#61;&#45;&#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=\"157\" style=\"vertical-align: -28px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836502030\">\n<div data-type=\"problem\" id=\"fs-id1167833338855\">\n<p id=\"fs-id1167836579407\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-acca5d104b42d14545f9a56cfe53892f_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;&#52;&#121;&#61;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#50;&#121;&#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=\"43\" width=\"124\" style=\"vertical-align: -17px;\" \/><\/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-c788abfeb3584aaf266b6912afedf2ab_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;&#50;&#121;&#45;&#122;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#120;&#43;&#121;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#120;&#43;&#52;&#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;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167824781184\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0b9645af99a3c5fa95bf361b5b7c2de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#92;&#92;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"178\" style=\"vertical-align: -17px;\" \/><\/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-f0f897cac49826afe141c9fddd2514ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"266\" style=\"vertical-align: -28px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829739049\">\n<div data-type=\"problem\" id=\"fs-id1167829714126\">\n<p id=\"fs-id1167833361725\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-658e02590102a038561b33995d315c33_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;&#121;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#61;&#121;&#43;&#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=\"43\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/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-ba24bf1b8454d1bcc9631d53803abc6d_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;&#45;&#52;&#122;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#43;&#50;&#121;&#43;&#50;&#122;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#53;&#121;&#43;&#55;&#122;&#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=\"166\" style=\"vertical-align: -28px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833009852\">\n<div data-type=\"problem\" id=\"fs-id1167829598294\">\n<p id=\"fs-id1167836364054\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cec1b4b035932aacb4d3ec601f5f2aa7_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;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#61;&#51;&#121;&#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=\"43\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/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-f3a9138d3800b9bb2529dcafeda94700_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;&#51;&#121;&#45;&#50;&#122;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#120;&#43;&#121;&#45;&#51;&#122;&#61;&#52;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#45;&#52;&#121;&#43;&#53;&#122;&#61;&#45;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"166\" style=\"vertical-align: -28px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836607370\">\n<p id=\"fs-id1167825830228\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5935af053965505cb4d22dd978b3cff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#92;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"178\" style=\"vertical-align: -17px;\" \/><\/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-28ee3346e08b9edfae671ecde1063a38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#92;&#32;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"266\" style=\"vertical-align: -28px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167833082484\">Write the system of equations that corresponds to the augmented matrix.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836391699\">\n<div data-type=\"problem\" id=\"fs-id1167829788432\">\n<p id=\"fs-id1167829787229\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b0e51ebbfc2996343863109bd32cfcd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#92;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"154\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829692032\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836706030\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-30308efe94cb7778038e20f86a3d3993_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#92;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"168\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836417217\">\n<p id=\"fs-id1167836542591\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-150c4b7b431e1fa3d74146eaec47f4ca_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;&#52;&#121;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#51;&#121;&#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=\"43\" width=\"124\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836448155\">\n<div data-type=\"problem\" id=\"fs-id1167833207785\">\n<p id=\"fs-id1167836626772\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6188da5685da8efa5f995d7e5647c86c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#92;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#92;&#92;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"243\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829832302\">\n<div data-type=\"problem\" id=\"fs-id1167836732801\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd4e64f834c333231f70fa0456151f25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#92;&#92;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#92;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"243\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836722460\">\n<p id=\"fs-id1167836627860\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-accc65103f160653cb0f6209ec02fd08_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;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#121;&#45;&#122;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#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=\"125\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829688535\"><strong data-effect=\"bold\">Use Row Operations on a Matrix<\/strong><\/p>\n<p id=\"fs-id1167836613644\">In the following exercises, perform the indicated operations on the augmented matrices.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836622536\">\n<div data-type=\"problem\" id=\"fs-id1167836287993\">\n<p id=\"fs-id1167836367083\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed1eae0684daf1533d5bcdc777df2926_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#92;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"154\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1167829690924\"><span class=\"token\">\u24d0<\/span> Interchange rows 1 and 2<\/p>\n<p id=\"fs-id1167836296936\"><span class=\"token\">\u24d1<\/span> Multiply row 2 by 3<\/p>\n<p id=\"fs-id1167836540243\"><span class=\"token\">\u24d2<\/span> Multiply row 2 by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> and add row 1 to it.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836538567\">\n<div data-type=\"problem\" id=\"fs-id1167829696382\">\n<p id=\"fs-id1167833014865\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d21c186b0756e4272e1e28177ee8fb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#92;&#92;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"168\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1167836650085\"><span class=\"token\">\u24d0<\/span> Interchange rows 1 and 2<\/p>\n<p id=\"fs-id1167836627418\"><span class=\"token\">\u24d1<\/span> Multiply row 1 by 4<\/p>\n<p id=\"fs-id1167836362828\"><span class=\"token\">\u24d2<\/span> Multiply row 2 by 3 and add row 1 to it.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836418067\">\n<p id=\"fs-id1167833054679\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1734f34ef5e5a5302d1e336e69f28dfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#92;&#92;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"178\" style=\"vertical-align: -17px;\" \/><\/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-e1c92e4cb87430721b809cefa8ab19b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#56;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#92;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"187\" style=\"vertical-align: -17px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af77a11d3e86e7094e4f078659914103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#56;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#92;&#32;&#50;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"196\" style=\"vertical-align: -17px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836706572\">\n<div data-type=\"problem\" id=\"fs-id1167829695986\">\n<p id=\"fs-id1167836625455\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ebefae081ef5da2fb2e9f33443e688be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#56;&#92;&#92;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#92;&#32;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#54;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"265\" style=\"vertical-align: -28px;\" \/><\/p>\n<p id=\"fs-id1167829620875\"><span class=\"token\">\u24d0<\/span> Interchange rows 2 and 3<\/p>\n<p><span class=\"token\">\u24d1<\/span> Multiply row 1 by 4<\/p>\n<p id=\"fs-id1167833361899\"><span class=\"token\">\u24d2<\/span> Multiply row 2 by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> and add to row 3.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829811863\">\n<div data-type=\"problem\" id=\"fs-id1167829752192\">\n<p id=\"fs-id1167833380181\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6eec96c6cad6b60519b86d44df745ba6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#92;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#92;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"243\" style=\"vertical-align: -28px;\" \/><\/p>\n<p id=\"fs-id1167829579826\"><span class=\"token\">\u24d0<\/span> Interchange rows 2 and 3<\/p>\n<p id=\"fs-id1167830077562\"><span class=\"token\">\u24d1<\/span> Multiply row 2 by 5<\/p>\n<p id=\"fs-id1167824704572\"><span class=\"token\">\u24d2<\/span> Multiply row 3 by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> and add to row 1.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167833021198\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-50e0652d81fc22b19fb52f72fdb2d6ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#54;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"253\" style=\"vertical-align: -28px;\" \/><\/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-ddb2f472e60186bad86b385f7c63c3b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#92;&#92;&#32;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#92;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"253\" style=\"vertical-align: -28px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49de49c4de7289424f681a5ea51d84f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#54;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#55;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#55;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"253\" style=\"vertical-align: -28px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824578322\">\n<div data-type=\"problem\" id=\"fs-id1167836626591\">\n<p id=\"fs-id1167836570727\">Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e76b9fb1a369e9b700bbe04e8689df6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"193\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167830123187\">\n<p id=\"fs-id1167824649012\">Perform the needed row operations that will get the first entry in both row 2 and row 3 to be zero in the augmented matrix: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc0583ba8c1aef311b7842aa7d96a1b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#92;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#92;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"255\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836693318\">\n<p id=\"fs-id1167832977161\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-add11e2e717c073d6d6b257b6f7bcd86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#92;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#55;&#92;&#92;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#55;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"261\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832982042\"><strong data-effect=\"bold\">Solve Systems of Equations Using Matrices<\/strong><\/p>\n<p id=\"fs-id1167836601156\">In the following exercises, solve each system of equations using a matrix.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836573324\">\n<div data-type=\"problem\" id=\"fs-id1167836684975\">\n<p id=\"fs-id1167836440482\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa1a43d8259c851b153cea4626b09906_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;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#121;&#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=\"43\" width=\"106\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833007373\">\n<div data-type=\"problem\" id=\"fs-id1167829683972\">\n<p id=\"fs-id1167836498172\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-613a98f81c16d9f595d171626e6e0341_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;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#121;&#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=\"43\" width=\"101\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836791945\">\n<p id=\"fs-id1167833019264\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3ac38dbb39343c28a60a287dfb114b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826170390\">\n<div data-type=\"problem\" id=\"fs-id1167836476890\">\n<p id=\"fs-id1167836476893\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e8273231db797c7100265f67cfab0c2d_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;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#50;&#121;&#61;&#45;&#50;&#92;&#92;&#32;&#120;&#43;&#121;&#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=\"43\" width=\"115\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833135703\">\n<div data-type=\"problem\" id=\"fs-id1167829689486\">\n<p id=\"fs-id1167829712505\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3e57e97f63d33c9e7d2bd31383fc1c06_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;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#120;&#43;&#51;&#121;&#61;&#51;&#92;&#92;&#32;&#120;&#43;&#51;&#121;&#61;&#49;&#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=\"43\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836486319\">\n<p id=\"fs-id1167836704893\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c4a2258b08828b82f5478b79177f57c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833139215\">In the following exercises, solve each system of equations using a matrix.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836609807\">\n<div data-type=\"problem\" id=\"fs-id1167836609809\">\n<p id=\"fs-id1167836611716\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8181316b3748decb74c54d021ebcb356_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;&#51;&#121;&#43;&#122;&#61;&#49;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#120;&#43;&#121;&#45;&#50;&#122;&#61;&#45;&#49;&#53;&#92;&#92;&#32;&#120;&#43;&#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=\"179\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836576451\">\n<div data-type=\"problem\" id=\"fs-id1167836539251\">\n<p id=\"fs-id1167836293793\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aeb7d7e3adf7e4362d6052229aaa09dc_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;&#121;&#43;&#51;&#122;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#50;&#121;&#45;&#122;&#61;&#49;&#48;&#92;&#92;&#32;&#120;&#43;&#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=\"157\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836629902\">\n<p id=\"fs-id1167836629904\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa0a6d07f515e53ab12ea2ded1b8efce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#53;&#44;&#50;&#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-id1167836635577\">\n<div data-type=\"problem\" id=\"fs-id1167836635580\">\n<p id=\"fs-id1167836627750\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8bd868c1694ce36a59f293f926eba24d_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;&#43;&#50;&#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<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836614234\">\n<div data-type=\"problem\" id=\"fs-id1167836614236\">\n<p id=\"fs-id1167836444948\"><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-id1167836573792\">\n<p id=\"fs-id1167836573794\"><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-id1167829850155\">\n<div data-type=\"problem\" id=\"fs-id1167829850157\">\n<p id=\"fs-id1167833047400\"><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-id1167836614876\">\n<div data-type=\"problem\" id=\"fs-id1167836323333\">\n<p id=\"fs-id1167836323335\"><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-id1167829650628\">\n<p id=\"fs-id1167829650630\"><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-id1167829756030\">\n<div data-type=\"problem\" id=\"fs-id1167829756032\">\n<p id=\"fs-id1167836376309\"><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-id1167833059137\">\n<div data-type=\"problem\" id=\"fs-id1167829694210\">\n<p id=\"fs-id1167829694212\"><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-id1167829686580\">\n<p id=\"fs-id1167829686583\"><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-id1167833047842\">\n<div data-type=\"problem\" id=\"fs-id1167833047844\">\n<p id=\"fs-id1167832945824\"><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-id1167836620917\">\n<div data-type=\"problem\" id=\"fs-id1167836299361\">\n<p id=\"fs-id1167836299363\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17a0af5d24d6d7a395581e79ab3804c9_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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#121;&#45;&#50;&#122;&#61;&#51;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"143\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836613179\">no solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836547142\">\n<div data-type=\"problem\" id=\"fs-id1167836547144\">\n<p id=\"fs-id1167836528457\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a0268b788043e6da4709aed42f991d6d_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"156\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836510303\">\n<div data-type=\"problem\" id=\"fs-id1167836689350\">\n<p id=\"fs-id1167836689352\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a926998830175c9ece4f51339239cd2_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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#120;&#43;&#51;&#121;&#45;&#55;&#122;&#61;&#49;&#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 data-type=\"solution\" id=\"fs-id1167836571645\">\n<p id=\"fs-id1167836433934\">no solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836480343\">\n<div data-type=\"problem\" id=\"fs-id1167836480345\">\n<p id=\"fs-id1167833380056\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e23646018d8fef5872140c164b2b30d8_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;&#50;&#122;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#121;&#43;&#51;&#122;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#120;&#43;&#51;&#121;&#45;&#54;&#122;&#61;&#49;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"174\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836531630\">\n<div data-type=\"problem\" id=\"fs-id1167836531632\">\n<p id=\"fs-id1167833056352\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b1d8d1fdcd73eb94555133db8290885_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;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#45;&#51;&#121;&#43;&#50;&#122;&#61;&#49;&#52;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#50;&#121;&#45;&#51;&#122;&#61;&#45;&#52;&#92;&#92;&#32;&#51;&#120;&#43;&#121;&#45;&#50;&#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=\"157\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836683544\">\n<p id=\"fs-id1167836683546\">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-e04660b4ec5bdce1c5f7b74c29cdb7af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#122;&#43;&#52;&#59;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#122;&#45;&#54;&#59;&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"192\" style=\"vertical-align: -6px;\" \/> is any real number<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833022411\">\n<div data-type=\"problem\" id=\"fs-id1167829719183\">\n<p id=\"fs-id1167829719185\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e470ee6d40da3b19ff6048685e8c899b_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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#50;&#121;&#61;&#49;&#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\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836516759\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d2378cf9ece07f88e615c7e4fc936822_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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#120;&#45;&#51;&#121;&#43;&#122;&#61;&#45;&#55;&#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-id1167836515229\">\n<p id=\"fs-id1167829833990\">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-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>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167833047439\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167833142303\">\n<div data-type=\"problem\" id=\"fs-id1167833142305\">\n<p id=\"fs-id1167836334886\">Solve the system of equations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8ad708fe8b845433e2833c8447d52e5_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;&#61;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#121;&#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=\"43\" width=\"102\" style=\"vertical-align: -17px;\" \/> <span class=\"token\">\u24d0<\/span> by graphing and <span class=\"token\">\u24d1<\/span> by substitution. <span class=\"token\">\u24d2<\/span> Which method do you prefer? Why?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829651171\">\n<div data-type=\"problem\" id=\"fs-id1167829651173\">\n<p id=\"fs-id1167829716737\">Solve the system of equations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-434f2a95b5a2042cd21886343bfa489d_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;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#61;&#121;&#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=\"43\" width=\"110\" style=\"vertical-align: -17px;\" \/> by substitution and explain all your steps in words.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826206613\">\n<p id=\"fs-id1167826206615\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836319947\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167833129255\"><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-id1167832994494\" data-alt=\"This table has 4 columns 5 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 column has the following statements: Write the augmented matrix for a system of equations, Use row operations on a matrix, Solve systems of equations using matrices, Write the augmented matrix for a system of equations, Use row operations on a matrix. 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_05_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns 5 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 column has the following statements: Write the augmented matrix for a system of equations, Use row operations on a matrix, Solve systems of equations using matrices, Write the augmented matrix for a system of equations, Use row operations on a matrix. The remaining columns are blank.\" \/><\/span><\/p>\n<p><span class=\"token\">\u24d1<\/span> After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?<\/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-id1167832999650\">\n<dt>matrix<\/dt>\n<dd id=\"fs-id1167836282956\">A matrix is a rectangular array of numbers arranged in rows and columns.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167836289514\">\n<dt>row-echelon form<\/dt>\n<dd id=\"fs-id1167836510587\">A matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2574","chapter","type-chapter","status-publish","hentry"],"part":2305,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2574","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":1,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2574\/revisions"}],"predecessor-version":[{"id":15240,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2574\/revisions\/15240"}],"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\/2574\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=2574"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=2574"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=2574"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=2574"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}