{"id":137,"date":"2025-02-13T22:44:12","date_gmt":"2025-02-13T22:44:12","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/solving-systems-with-gaussian-elimination\/"},"modified":"2026-03-24T16:22:03","modified_gmt":"2026-03-24T16:22:03","slug":"solving-systems-with-gaussian-elimination","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/solving-systems-with-gaussian-elimination\/","title":{"raw":"Solving Systems with Gaussian Elimination: Learn It 1","rendered":"Solving Systems with Gaussian Elimination: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Convert between augmented matrices and systems of equations<\/li>\r\n \t<li>Perform row operations on a matrix.<\/li>\r\n \t<li>Solve a system of linear equations using row operations.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Augmented Matrix of a System of Equations<\/h2>\r\nA matrix can serve as a device for representing and solving a system of equations. To express a system in matrix form, we extract the coefficients of the variables and the constants, and these become the entries of the matrix. We use a vertical line to separate the coefficient entries from the constants, essentially replacing the equal signs. When a system is written in this form, we call it an <strong>augmented matrix<\/strong>.\r\n\r\n<img class=\"aligncenter wp-image-5025\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03184054\/10.2.L.1.Diagram-300x98.png\" alt=\"A diagram showing a system of two linear equations being rewritten as an augmented matrix. The coefficients of x form the first column, the coefficients of y form the second column, and the constants form the third column.\" width=\"508\" height=\"166\" \/>\r\n<ul>\r\n \t<li>Each <strong>column<\/strong> then would be the coefficients of one of the variables in the system or the constants.<\/li>\r\n \t<li>A <strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">vertical line<\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"> replaces the equal signs.<\/span><\/li>\r\n<\/ul>\r\nWe call the resulting matrix the <strong>augmented<\/strong> matrix for the system of equations.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">For example, consider the following [latex]2\\times 2[\/latex] system of equations.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}3x+4y=7\\\\ 4x - 2y=5\\end{array}[\/latex]<\/div>\r\nWe can write this system as an augmented matrix:\r\n<div style=\"text-align: center;\">[latex]\\left[\\begin{array}{cc|c}\\hfill 3&amp; \\hfill 4&amp; \\hfill 7\\\\ \\hfill 4&amp; \\hfill -2&amp; \\hfill 5\\\\ \\end{array}\\right][\/latex]<\/div>\r\nWe can also write a matrix containing just the coefficients. This is called the <strong>coefficient matrix<\/strong>.\r\n<div style=\"text-align: center;\">[latex]\\left[\\begin{array}{cc}3&amp; 4\\\\ 4&amp; -2\\end{array}\\right][\/latex]<\/div>\r\nA three-by-three <strong>system of equations<\/strong> such as\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}3x-y-z=0\\hfill \\\\ \\text{ }x+y=5\\hfill \\\\ \\text{ }2x - 3z=2\\hfill \\end{array}[\/latex]<\/div>\r\nhas a coefficient matrix\r\n<div style=\"text-align: center;\">[latex]\\left[\\begin{array}{rrr}\\hfill 3&amp; \\hfill -1&amp; \\hfill -1\\\\ \\hfill 1&amp; \\hfill 1&amp; \\hfill 0\\\\ \\hfill 2&amp; \\hfill 0&amp; \\hfill -3\\end{array}\\right][\/latex]<\/div>\r\nand is represented by the augmented matrix\r\n<p style=\"text-align: center;\">[latex]\\left[\\begin{array}{ccc|c}\\hfill 3&amp; \\hfill -1&amp; \\hfill -1&amp; \\hfill 0\\\\ \\hfill 1&amp; \\hfill 1&amp; \\hfill 0&amp; \\hfill 5\\\\ \\hfill 2&amp; \\hfill 0&amp; \\hfill -3&amp; \\hfill 2\\\\ \\end{array}\\right][\/latex]<\/p>\r\nNotice that the matrix is written so that the variables line up in their own columns: [latex]x-[\/latex]terms go in the first column, [latex]y-[\/latex]terms in the second column, and [latex]z-[\/latex]terms in the third column. It is very important that each equation is written in standard form [latex]ax+by+cz=d[\/latex] so that the variables line up. When there is a missing variable term in an equation, the coefficient is [latex]0[\/latex].\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a system of equations, write an augmented matrix<\/strong>\r\n<ol>\r\n \t<li>Write the coefficients of the [latex]x-[\/latex]terms as the numbers down the first column.<\/li>\r\n \t<li>Write the coefficients of the [latex]y-[\/latex]terms as the numbers down the second column.<\/li>\r\n \t<li>If there are [latex]z-[\/latex]terms, write the coefficients as the numbers down the third column.<\/li>\r\n \t<li>Draw a vertical line and write the constants to the right of the line.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Write the augmented matrix for the given system of equations.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{ }x+2y-z=3\\hfill \\\\ \\text{ }2x-y+2z=6\\hfill \\\\ \\text{ }x - 3y+3z=4\\hfill \\end{array}[\/latex]<\/div>\r\n[reveal-answer q=\"863191\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"863191\"]\r\n\r\nThe augmented matrix displays the coefficients of the variables and an additional column for the constants.\r\n<div style=\"text-align: center;\">[latex]\\left[\\begin{array}{ccc|c}\\hfill 1&amp; \\hfill 2&amp; \\hfill -1&amp; \\hfill 3\\\\ \\hfill 2&amp; \\hfill -1&amp; \\hfill 2&amp; \\hfill 6\\\\ \\hfill 1&amp; \\hfill -3&amp; \\hfill 3&amp; \\hfill 4\\\\ \\end{array}\\right][\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321744[\/ohm_question]<\/section><section aria-label=\"Try It\"><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321745[\/ohm_question]<\/section><\/section>We can use augmented matrices to help us solve systems of equations because they simplify operations when the systems are not encumbered by the variables. However, it is important to understand how to move back and forth between formats in order to make finding solutions smoother and more intuitive. Here, we will use the information in an augmented matrix to write the <strong>system of equations<\/strong> in standard form.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the system of equations from the augmented matrix.\r\n<div style=\"text-align: center;\">[latex]\\left[\\begin{array}{ccc|c}\\hfill 1&amp; \\hfill -3&amp; \\hfill -5&amp; \\hfill -2\\\\ \\hfill 2&amp; \\hfill -5&amp; \\hfill -4&amp; \\hfill 5\\\\ \\hfill -3&amp; \\hfill 5&amp; \\hfill 4&amp; \\hfill 6\\\\ \\end{array}\\right][\/latex]<\/div>\r\n[reveal-answer q=\"811290\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"811290\"]\r\nWhen the columns represent the variables [latex]x[\/latex], [latex]y[\/latex], and [latex]z[\/latex],\r\n<div style=\"text-align: center;\">[latex]\\left[\\begin{array}{ccc|c}\\hfill 1&amp; \\hfill -3&amp; \\hfill -5&amp; \\hfill -2\\\\ \\hfill 2&amp; \\hfill -5&amp; \\hfill -4&amp; \\hfill 5\\\\ \\hfill -3&amp; \\hfill 5&amp; \\hfill 4&amp; \\hfill 6\\\\ \\end{array}\\right]\\to \\begin{array}{l}x - 3y - 5z=-2\\hfill \\\\ 2x - 5y - 4z=5\\hfill \\\\ -3x+5y+4z=6\\hfill \\end{array}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321746[\/ohm_question]<\/section><\/div>\r\n<dl id=\"fs-id1165137456553\" class=\"definition\">\r\n \t<dd id=\"fs-id1165132961357\"><\/dd>\r\n<\/dl>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Convert between augmented matrices and systems of equations<\/li>\n<li>Perform row operations on a matrix.<\/li>\n<li>Solve a system of linear equations using row operations.<\/li>\n<\/ul>\n<\/section>\n<h2>Augmented Matrix of a System of Equations<\/h2>\n<p>A matrix can serve as a device for representing and solving a system of equations. To express a system in matrix form, we extract the coefficients of the variables and the constants, and these become the entries of the matrix. We use a vertical line to separate the coefficient entries from the constants, essentially replacing the equal signs. When a system is written in this form, we call it an <strong>augmented matrix<\/strong>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5025\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03184054\/10.2.L.1.Diagram-300x98.png\" alt=\"A diagram showing a system of two linear equations being rewritten as an augmented matrix. The coefficients of x form the first column, the coefficients of y form the second column, and the constants form the third column.\" width=\"508\" height=\"166\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03184054\/10.2.L.1.Diagram-300x98.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03184054\/10.2.L.1.Diagram-65x21.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03184054\/10.2.L.1.Diagram-225x74.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03184054\/10.2.L.1.Diagram-350x115.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03184054\/10.2.L.1.Diagram.png 644w\" sizes=\"(max-width: 508px) 100vw, 508px\" \/><\/p>\n<ul>\n<li>Each <strong>column<\/strong> then would be the coefficients of one of the variables in the system or the constants.<\/li>\n<li>A <strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">vertical line<\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"> replaces the equal signs.<\/span><\/li>\n<\/ul>\n<p>We call the resulting matrix the <strong>augmented<\/strong> matrix for the system of equations.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">For example, consider the following [latex]2\\times 2[\/latex] system of equations.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}3x+4y=7\\\\ 4x - 2y=5\\end{array}[\/latex]<\/div>\n<p>We can write this system as an augmented matrix:<\/p>\n<div style=\"text-align: center;\">[latex]\\left[\\begin{array}{cc|c}\\hfill 3& \\hfill 4& \\hfill 7\\\\ \\hfill 4& \\hfill -2& \\hfill 5\\\\ \\end{array}\\right][\/latex]<\/div>\n<p>We can also write a matrix containing just the coefficients. This is called the <strong>coefficient matrix<\/strong>.<\/p>\n<div style=\"text-align: center;\">[latex]\\left[\\begin{array}{cc}3& 4\\\\ 4& -2\\end{array}\\right][\/latex]<\/div>\n<p>A three-by-three <strong>system of equations<\/strong> such as<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}3x-y-z=0\\hfill \\\\ \\text{ }x+y=5\\hfill \\\\ \\text{ }2x - 3z=2\\hfill \\end{array}[\/latex]<\/div>\n<p>has a coefficient matrix<\/p>\n<div style=\"text-align: center;\">[latex]\\left[\\begin{array}{rrr}\\hfill 3& \\hfill -1& \\hfill -1\\\\ \\hfill 1& \\hfill 1& \\hfill 0\\\\ \\hfill 2& \\hfill 0& \\hfill -3\\end{array}\\right][\/latex]<\/div>\n<p>and is represented by the augmented matrix<\/p>\n<p style=\"text-align: center;\">[latex]\\left[\\begin{array}{ccc|c}\\hfill 3& \\hfill -1& \\hfill -1& \\hfill 0\\\\ \\hfill 1& \\hfill 1& \\hfill 0& \\hfill 5\\\\ \\hfill 2& \\hfill 0& \\hfill -3& \\hfill 2\\\\ \\end{array}\\right][\/latex]<\/p>\n<p>Notice that the matrix is written so that the variables line up in their own columns: [latex]x-[\/latex]terms go in the first column, [latex]y-[\/latex]terms in the second column, and [latex]z-[\/latex]terms in the third column. It is very important that each equation is written in standard form [latex]ax+by+cz=d[\/latex] so that the variables line up. When there is a missing variable term in an equation, the coefficient is [latex]0[\/latex].<\/p>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a system of equations, write an augmented matrix<\/strong><\/p>\n<ol>\n<li>Write the coefficients of the [latex]x-[\/latex]terms as the numbers down the first column.<\/li>\n<li>Write the coefficients of the [latex]y-[\/latex]terms as the numbers down the second column.<\/li>\n<li>If there are [latex]z-[\/latex]terms, write the coefficients as the numbers down the third column.<\/li>\n<li>Draw a vertical line and write the constants to the right of the line.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Write the augmented matrix for the given system of equations.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{ }x+2y-z=3\\hfill \\\\ \\text{ }2x-y+2z=6\\hfill \\\\ \\text{ }x - 3y+3z=4\\hfill \\end{array}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q863191\">Show Solution<\/button><\/p>\n<div id=\"q863191\" class=\"hidden-answer\" style=\"display: none\">\n<p>The augmented matrix displays the coefficients of the variables and an additional column for the constants.<\/p>\n<div style=\"text-align: center;\">[latex]\\left[\\begin{array}{ccc|c}\\hfill 1& \\hfill 2& \\hfill -1& \\hfill 3\\\\ \\hfill 2& \\hfill -1& \\hfill 2& \\hfill 6\\\\ \\hfill 1& \\hfill -3& \\hfill 3& \\hfill 4\\\\ \\end{array}\\right][\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321744\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321744&theme=lumen&iframe_resize_id=ohm321744&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section aria-label=\"Try It\">\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321745\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321745&theme=lumen&iframe_resize_id=ohm321745&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n<p>We can use augmented matrices to help us solve systems of equations because they simplify operations when the systems are not encumbered by the variables. However, it is important to understand how to move back and forth between formats in order to make finding solutions smoother and more intuitive. Here, we will use the information in an augmented matrix to write the <strong>system of equations<\/strong> in standard form.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Find the system of equations from the augmented matrix.<\/p>\n<div style=\"text-align: center;\">[latex]\\left[\\begin{array}{ccc|c}\\hfill 1& \\hfill -3& \\hfill -5& \\hfill -2\\\\ \\hfill 2& \\hfill -5& \\hfill -4& \\hfill 5\\\\ \\hfill -3& \\hfill 5& \\hfill 4& \\hfill 6\\\\ \\end{array}\\right][\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q811290\">Show Solution<\/button><\/p>\n<div id=\"q811290\" class=\"hidden-answer\" style=\"display: none\">\nWhen the columns represent the variables [latex]x[\/latex], [latex]y[\/latex], and [latex]z[\/latex],<\/p>\n<div style=\"text-align: center;\">[latex]\\left[\\begin{array}{ccc|c}\\hfill 1& \\hfill -3& \\hfill -5& \\hfill -2\\\\ \\hfill 2& \\hfill -5& \\hfill -4& \\hfill 5\\\\ \\hfill -3& \\hfill 5& \\hfill 4& \\hfill 6\\\\ \\end{array}\\right]\\to \\begin{array}{l}x - 3y - 5z=-2\\hfill \\\\ 2x - 5y - 4z=5\\hfill \\\\ -3x+5y+4z=6\\hfill \\end{array}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321746\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321746&theme=lumen&iframe_resize_id=ohm321746&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n<dl id=\"fs-id1165137456553\" class=\"definition\">\n<dd id=\"fs-id1165132961357\"><\/dd>\n<\/dl>\n","protected":false},"author":6,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":514,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/137"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/137\/revisions"}],"predecessor-version":[{"id":5981,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/137\/revisions\/5981"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/514"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/137\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=137"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=137"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=137"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=137"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}