{"id":1471,"date":"2025-07-25T02:04:15","date_gmt":"2025-07-25T02:04:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1471"},"modified":"2026-03-24T07:17:47","modified_gmt":"2026-03-24T07:17:47","slug":"solving-systems-with-gaussian-elimination-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/solving-systems-with-gaussian-elimination-fresh-take\/","title":{"raw":"Solving Systems with Gaussian Elimination: Fresh Take","rendered":"Solving Systems with Gaussian Elimination: Fresh Take"},"content":{"raw":"<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\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Definition<\/strong>: An augmented matrix is a way to represent a system of linear equations in matrix form.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Structure<\/strong>:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Coefficients of variables form the main part of the matrix<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Constants are separated by a vertical line<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each row represents one equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each column (before the line) represents coefficients of one variable<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Coefficient Matrix<\/strong>: The matrix containing only the coefficients of variables (without the constants)<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Importance of Standard Form<\/strong>: Equations should be in the form [latex]ax + by + cz = d[\/latex] for proper alignment in the matrix<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Zero Coefficients<\/strong>: When a variable is missing from an equation, its coefficient is represented as [latex]0[\/latex] in the matrix<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Write the augmented matrix of the given system of equations.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}4x - 3y=11\\\\ 3x+2y=4\\end{array}[\/latex]<\/div>\r\n<div><\/div>\r\n<div>[reveal-answer q=\"769522\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"769522\"][latex]\\left[\\begin{array}{cc|c}\\hfill 4&amp; \\hfill -3&amp; \\hfill 11\\\\ \\hfill 3&amp; \\hfill 2&amp; \\hfill 4\\\\ \\end{array}\\right][\/latex][\/hidden-answer]<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Write 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 -1&amp; \\hfill 1&amp; \\hfill 5\\\\ \\hfill 2&amp; \\hfill -1&amp; \\hfill 3&amp; \\hfill 1\\\\ \\hfill 0&amp; \\hfill 1&amp; \\hfill 1&amp; \\hfill -9\\\\ \\end{array}\\right][\/latex]<\/div>\r\n<div>[reveal-answer q=\"696438\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"696438\"][latex]\\begin{array}{c}x-y+z=5\\\\ 2x-y+3z=1\\\\ y+z=-9\\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/section>\r\n<h2>Performing Row Operations on a Matrix<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Row Operations<\/strong>: Transformations applied to matrices that preserve the solution set of the corresponding system of equations.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Types of Row Operations<\/strong>:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Interchanging any two rows<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiplying a row by a non-zero constant<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Adding a multiple of one row to another row<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Row-Echelon Form<\/strong>: A matrix form where:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">The first non-zero element in each row (leading entry) is 1<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each leading 1 is to the right of the leading 1 in the row above<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rows with all zero elements are at the bottom<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Gaussian Elimination<\/strong>: A method to transform a matrix into row-echelon form using row operations.<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Process of Gaussian Elimination<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Start with the leftmost column<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find a non-zero entry in this column (if all zero, move to next column)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Move the row with this non-zero entry to the top (if not already there)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Make this entry a 1 by dividing the row by the entry's value<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use this 1 to eliminate all other entries in this column<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Repeat steps 1-5 for the next column, working only on rows below the current row<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Write the system of equations in row-echelon form.[latex]\\begin{array}{l}\\text{ }x - 2y+3z=9\\hfill \\\\ \\text{ }-x+3y=-4\\hfill \\\\ 2x - 5y+5z=17\\hfill \\end{array}[\/latex]\r\n<p style=\"text-align: left;\">[reveal-answer q=\"971348\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"971348\"]\r\n[latex]\\left[\\begin{array}{ccc|c}\\hfill 1&amp; \\hfill -\\frac{5}{2}&amp; \\hfill \\frac{5}{2}&amp; \\hfill \\frac{17}{2}\\\\ \\hfill 0&amp; \\hfill 1&amp; \\hfill 5&amp; \\hfill 9\\\\ \\hfill 0&amp; \\hfill 0&amp; \\hfill 1&amp; \\hfill 2\\\\ \\end{array}\\right][\/latex]\r\n[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve the given system by Gaussian elimination.\r\n<div>[latex]\\begin{array}{l}4x+3y=11\\hfill \\\\ \\text{ }\\text{}\\text{}x - 3y=-1\\hfill \\end{array}[\/latex]<\/div>\r\n<div>[reveal-answer q=\"761791\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"761791\"][latex]\\left(2,1\\right)[\/latex][\/hidden-answer]<\/div>\r\n<\/section>\r\n<div><section class=\"textbox example\" aria-label=\"Example\">Solve the system using Gaussian Elimination.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}x+4y-z=4\\\\ 2x+5y+8z=15\\\\ x+3y - 3z=1\\end{array}[\/latex]<\/p>\r\n[reveal-answer q=\"589600\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"589600\"]\r\n\r\n[latex]\\left(1,1,1\\right)[\/latex][\/hidden-answer]\r\n\r\n<\/section><\/div>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bbcgccha-NgXXKmQHFDg\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/NgXXKmQHFDg?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bbcgccha-NgXXKmQHFDg\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851077&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bbcgccha-NgXXKmQHFDg&amp;vembed=0&amp;video_id=NgXXKmQHFDg&amp;video_target=tpm-plugin-bbcgccha-NgXXKmQHFDg\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+2+-+Solve+a+System+of+Two+Equations+with+Using+an+Augmented+Matrix+(Row+Echelon+Form)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Solve a System of Two Equations with Using an Augmented Matrix (Row Echelon Form)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hefbabfb-YPgq9K0R3Xw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/YPgq9K0R3Xw?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hefbabfb-YPgq9K0R3Xw\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851078&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hefbabfb-YPgq9K0R3Xw&amp;vembed=0&amp;video_id=YPgq9K0R3Xw&amp;video_target=tpm-plugin-hefbabfb-YPgq9K0R3Xw\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+2+-+Solve+a+System+of+Three+Equations+with+Using+an+Augmented+Matrix+(REF+-+no+solution))_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Solve a System of Three Equations with Using an Augmented Matrix (REF - no solution))\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<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<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\"><strong>Definition<\/strong>: An augmented matrix is a way to represent a system of linear equations in matrix form.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Structure<\/strong>:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Coefficients of variables form the main part of the matrix<\/li>\n<li class=\"whitespace-normal break-words\">Constants are separated by a vertical line<\/li>\n<li class=\"whitespace-normal break-words\">Each row represents one equation<\/li>\n<li class=\"whitespace-normal break-words\">Each column (before the line) represents coefficients of one variable<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Coefficient Matrix<\/strong>: The matrix containing only the coefficients of variables (without the constants)<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Importance of Standard Form<\/strong>: Equations should be in the form [latex]ax + by + cz = d[\/latex] for proper alignment in the matrix<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Zero Coefficients<\/strong>: When a variable is missing from an equation, its coefficient is represented as [latex]0[\/latex] in the matrix<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Write the augmented matrix of the given system of equations.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}4x - 3y=11\\\\ 3x+2y=4\\end{array}[\/latex]<\/div>\n<div><\/div>\n<div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q769522\">Show Solution<\/button><\/p>\n<div id=\"q769522\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left[\\begin{array}{cc|c}\\hfill 4& \\hfill -3& \\hfill 11\\\\ \\hfill 3& \\hfill 2& \\hfill 4\\\\ \\end{array}\\right][\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Write the system of equations from the augmented matrix.<\/p>\n<div style=\"text-align: center;\">[latex]\\left[\\begin{array}{ccc|c}\\hfill 1& \\hfill -1& \\hfill 1& \\hfill 5\\\\ \\hfill 2& \\hfill -1& \\hfill 3& \\hfill 1\\\\ \\hfill 0& \\hfill 1& \\hfill 1& \\hfill -9\\\\ \\end{array}\\right][\/latex]<\/div>\n<div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q696438\">Show Solution<\/button><\/p>\n<div id=\"q696438\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{array}{c}x-y+z=5\\\\ 2x-y+3z=1\\\\ y+z=-9\\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2>Performing Row Operations on a Matrix<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\"><strong>Row Operations<\/strong>: Transformations applied to matrices that preserve the solution set of the corresponding system of equations.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Types of Row Operations<\/strong>:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Interchanging any two rows<\/li>\n<li class=\"whitespace-normal break-words\">Multiplying a row by a non-zero constant<\/li>\n<li class=\"whitespace-normal break-words\">Adding a multiple of one row to another row<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Row-Echelon Form<\/strong>: A matrix form where:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The first non-zero element in each row (leading entry) is 1<\/li>\n<li class=\"whitespace-normal break-words\">Each leading 1 is to the right of the leading 1 in the row above<\/li>\n<li class=\"whitespace-normal break-words\">Rows with all zero elements are at the bottom<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Gaussian Elimination<\/strong>: A method to transform a matrix into row-echelon form using row operations.<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Process of Gaussian Elimination<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Start with the leftmost column<\/li>\n<li class=\"whitespace-normal break-words\">Find a non-zero entry in this column (if all zero, move to next column)<\/li>\n<li class=\"whitespace-normal break-words\">Move the row with this non-zero entry to the top (if not already there)<\/li>\n<li class=\"whitespace-normal break-words\">Make this entry a 1 by dividing the row by the entry&#8217;s value<\/li>\n<li class=\"whitespace-normal break-words\">Use this 1 to eliminate all other entries in this column<\/li>\n<li class=\"whitespace-normal break-words\">Repeat steps 1-5 for the next column, working only on rows below the current row<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Write the system of equations in row-echelon form.[latex]\\begin{array}{l}\\text{ }x - 2y+3z=9\\hfill \\\\ \\text{ }-x+3y=-4\\hfill \\\\ 2x - 5y+5z=17\\hfill \\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q971348\">Show Solution<\/button><\/p>\n<div id=\"q971348\" class=\"hidden-answer\" style=\"display: none\">\n[latex]\\left[\\begin{array}{ccc|c}\\hfill 1& \\hfill -\\frac{5}{2}& \\hfill \\frac{5}{2}& \\hfill \\frac{17}{2}\\\\ \\hfill 0& \\hfill 1& \\hfill 5& \\hfill 9\\\\ \\hfill 0& \\hfill 0& \\hfill 1& \\hfill 2\\\\ \\end{array}\\right][\/latex]\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the given system by Gaussian elimination.<\/p>\n<div>[latex]\\begin{array}{l}4x+3y=11\\hfill \\\\ \\text{ }\\text{}\\text{}x - 3y=-1\\hfill \\end{array}[\/latex]<\/div>\n<div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q761791\">Show Solution<\/button><\/p>\n<div id=\"q761791\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left(2,1\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the system using Gaussian Elimination.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}x+4y-z=4\\\\ 2x+5y+8z=15\\\\ x+3y - 3z=1\\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q589600\">Show Solution<\/button><\/p>\n<div id=\"q589600\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(1,1,1\\right)[\/latex]<\/p><\/div>\n<\/div>\n<\/section>\n<\/div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bbcgccha-NgXXKmQHFDg\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/NgXXKmQHFDg?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bbcgccha-NgXXKmQHFDg\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851077&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bbcgccha-NgXXKmQHFDg&amp;vembed=0&amp;video_id=NgXXKmQHFDg&amp;video_target=tpm-plugin-bbcgccha-NgXXKmQHFDg\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+2+-+Solve+a+System+of+Two+Equations+with+Using+an+Augmented+Matrix+(Row+Echelon+Form)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Solve a System of Two Equations with Using an Augmented Matrix (Row Echelon Form)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hefbabfb-YPgq9K0R3Xw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/YPgq9K0R3Xw?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hefbabfb-YPgq9K0R3Xw\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851078&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hefbabfb-YPgq9K0R3Xw&amp;vembed=0&amp;video_id=YPgq9K0R3Xw&amp;video_target=tpm-plugin-hefbabfb-YPgq9K0R3Xw\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+2+-+Solve+a+System+of+Three+Equations+with+Using+an+Augmented+Matrix+(REF+-+no+solution))_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Solve a System of Three Equations with Using an Augmented Matrix (REF &#8211; no solution))\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":14,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex 2: Solve a System of Two Equations with Using an Augmented Matrix (Row Echelon Form)\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/NgXXKmQHFDg\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 2: Solve a System of Three Equations with Using an Augmented Matrix (REF - no solution))\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/YPgq9K0R3Xw\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":514,"module-header":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"Ex 2: Solve a System of Two Equations with Using an Augmented Matrix (Row Echelon Form)","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/NgXXKmQHFDg","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Ex 2: Solve a System of Three Equations with Using an Augmented Matrix (REF - no solution))","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/YPgq9K0R3Xw","project":"","license":"arr","license_terms":"Standard YouTube License"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12851077&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bbcgccha-NgXXKmQHFDg&vembed=0&video_id=NgXXKmQHFDg&video_target=tpm-plugin-bbcgccha-NgXXKmQHFDg'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12851078&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hefbabfb-YPgq9K0R3Xw&vembed=0&video_id=YPgq9K0R3Xw&video_target=tpm-plugin-hefbabfb-YPgq9K0R3Xw'><\/script>\n","media_targets":["tpm-plugin-bbcgccha-NgXXKmQHFDg","tpm-plugin-hefbabfb-YPgq9K0R3Xw"]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1471"}],"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\/67"}],"version-history":[{"count":2,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1471\/revisions"}],"predecessor-version":[{"id":3385,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1471\/revisions\/3385"}],"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\/1471\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1471"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1471"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1471"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1471"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}