{"id":1464,"date":"2025-07-25T01:59:09","date_gmt":"2025-07-25T01:59:09","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1464"},"modified":"2026-03-24T07:24:34","modified_gmt":"2026-03-24T07:24:34","slug":"systems-of-linear-equations-three-variables-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/systems-of-linear-equations-three-variables-fresh-take\/","title":{"raw":"Systems of Linear Equations: Three Variables: Fresh Take","rendered":"Systems of Linear Equations: Three Variables: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Solve systems of three equations in three variables.<\/li>\r\n \t<li>Identify inconsistent systems of equations containing three variables.<\/li>\r\n \t<li>Express the solution of a system of dependent equations containing three variables.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Systems of Three Equations in Three Variables<\/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\">System Structure\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Each equation has form [latex]ax + by + cz = d[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Three equations with three unknowns ([latex]x[\/latex], [latex]y[\/latex], [latex]z[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each equation represents a plane in 3D space<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Types of Solutions\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">One solution: [latex]{(x,y,z)}[\/latex] (intersection point of three planes)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">No solution (inconsistent): leads to contradiction like [latex]3=0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Infinite solutions (dependent): leads to identity like [latex]0=0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 aria-label=\"Try It\">Solve Systems of Three Equations in Three Variables<\/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\">Step-by-Step Solution Process\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Choose any two equations to work with first<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Eliminate one variable to create a two-variable equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Repeat with another pair to get a second two-variable equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve the resulting two-by-two system<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use back-substitution to find the final variable<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Back-Substitution Strategy\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Once you find one variable, plug it into simpler equations<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Work from simplest to most complex equations<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Keep track of positive\/negative signs carefully<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Variable Selection\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Choose the variable that's easiest to eliminate first<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Look for equations where variables are missing<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Consider coefficients that make elimination easier<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solution Verification\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Always check solution in ALL original equations<\/li>\r\n \t<li class=\"whitespace-normal break-words\">A true solution works in every equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">One equation being false means the solution is incorrect<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Solve the following system of equations:\r\n<center>[latex]\\begin{align*}\r\n2x + y - 3z &amp;= 4 \\\r\nx - y + z &amp;= 2 \\\r\n3x + 2y &amp;= 8\r\n\\end{align*}[\/latex]<\/center>\r\n[reveal-answer q=\"428907\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"428907\"]\r\n[latex]x= \\frac{9}{4}, y= \\frac{5}{8}, z= \\frac{3}{8}[\/latex]\r\n[\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video, we show another example of using back-substitution to solve a system in three variables.\r\n<script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ededbedc-HHIjTChrIxE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/HHIjTChrIxE?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ededbedc-HHIjTChrIxE\" 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=12850729&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ededbedc-HHIjTChrIxE&amp;vembed=0&amp;video_id=HHIjTChrIxE&amp;video_target=tpm-plugin-ededbedc-HHIjTChrIxE\" 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+-+Solve+a+System+of+3+Equations+with+3+Unknowns+Using+Back+Substitution_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solve a System of 3 Equations with 3 Unknowns Using Back Substitution\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Solving a System of Three Equations in Three Variables by Elimination<\/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\">Preparation Steps\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Write all equations in standard form: [latex]ax + by + cz = d[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Clear any fractions by multiplying through<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Label equations (1), (2), (3) for tracking<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Plan which variable to eliminate first<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Elimination Process\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Choose same variable to eliminate from two pairs of equations<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Create two new equations with two variables<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve resulting two-by-two system<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Back-substitute to find final variable<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Strategy\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Look for equations where coefficients are already opposites<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Choose variable that's easiest to eliminate<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Keep coefficients as simple as possible<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Create upper triangular form when possible<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Systematic Approach\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Always follow elimination steps in order<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Track equations with numbers\/labels<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Show all work clearly<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Verify solution in all original equations<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Solve the system of equations in three variables.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}2x+y - 2z=-1\\hfill \\\\ 3x - 3y-z=5\\hfill \\\\ x - 2y+3z=6\\hfill \\end{array}[\/latex]<\/p>\r\n[reveal-answer q=\"232738\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"232738\"]\r\n\r\n[latex]\\left(1,-1,1\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following videos for more examples of the algebra you may encounter when solving systems with three variables.\r\n<script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ehhecaff-r6htz3gaHZ0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/r6htz3gaHZ0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ehhecaff-r6htz3gaHZ0\" 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=12850730&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ehhecaff-r6htz3gaHZ0&amp;vembed=0&amp;video_id=r6htz3gaHZ0&amp;video_target=tpm-plugin-ehhecaff-r6htz3gaHZ0\" 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+-+System+of+Three+Equations+with+Three+Unknowns+Using+Elimination_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: System of Three Equations with Three Unknowns Using Elimination\u201d here (opens in new window).<\/a>\r\n\r\n<script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-egffghab-3RbVSvvRyeI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/3RbVSvvRyeI?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-egffghab-3RbVSvvRyeI\" 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=12850731&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-egffghab-3RbVSvvRyeI&amp;vembed=0&amp;video_id=3RbVSvvRyeI&amp;video_target=tpm-plugin-egffghab-3RbVSvvRyeI\" 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+1+-+System+of+Three+Equations+with+Three+Unknowns+Using+Elimination_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: System of Three Equations with Three Unknowns Using Elimination\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video,\u00a0you will see a visual representation of the three possible outcomes for solutions to a system of equations in three variables. There is also a worked example of solving a system using elimination.\r\n<script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-eabhhbcd-wIE8KSpb-E8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/wIE8KSpb-E8?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-eabhhbcd-wIE8KSpb-E8\" 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=12850732&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-eabhhbcd-wIE8KSpb-E8&amp;vembed=0&amp;video_id=wIE8KSpb-E8&amp;video_target=tpm-plugin-eabhhbcd-wIE8KSpb-E8\" 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\/Systems+of+Equations+in+Three+Variables+-+Part+1+of+2_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSystems of Equations in Three Variables: Part 1 of 2\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Inconsistent Systems of Equations Containing Three Variables<\/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\">Key Characteristics:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">No solution exists that satisfies all equations<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Results in a contradiction (e.g., [latex]0 = 5[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Represents planes that don't intersect at a common point<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Geometric Interpretations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Three parallel planes<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Two parallel planes with one intersecting plane<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Three planes intersecting in different locations<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Detection Method:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Elimination process leads to a contradiction<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Often takes several steps to reveal the contradiction<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Cannot be determined by looking at equations alone<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Solve the system of three equations in three variables.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{ }x+y+z=2\\hfill \\\\ \\text{ }y - 3z=1\\hfill \\\\ 2x+y+5z=0\\hfill \\end{array}[\/latex]<\/p>\r\n[reveal-answer q=\"563392\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"563392\"]\r\n\r\nNo solution.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the video below for another example of using elimination in a system that has no solution.\r\n<script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-aeghaacc-ryNQsWrUoJw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ryNQsWrUoJw?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-aeghaacc-ryNQsWrUoJw\" 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=12850733&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-aeghaacc-ryNQsWrUoJw&amp;vembed=0&amp;video_id=ryNQsWrUoJw&amp;video_target=tpm-plugin-aeghaacc-ryNQsWrUoJw\" 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+4+-+System+of+Three+Equations+with+Three+Unknowns+Using+Elimination+(No+Solution)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 4: System of Three Equations with Three Unknowns Using Elimination (No Solution)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Dependent Systems of Equations Containing Three Variables<\/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\">Key Characteristics:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Infinite solutions exist<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Results in an identity (e.g., [latex]0 = 0[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Can express solution using one variable<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Geometric Interpretations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Three identical planes<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Two identical planes intersecting a third<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Three planes intersecting along a line<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Solve the following system.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}x+y+z=7 \\\\ 3x - 2y-z=4 \\\\ x+6y+5z=24 \\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"195958\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"195958\"]\r\n\r\nInfinitely many number of solutions of the form [latex]\\left(x,4x - 11,-5x+18\\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">See the following video for another example of a dependent three-by-three system.\r\n<script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cedbhgaa-mThiwW8nYAU\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/mThiwW8nYAU?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cedbhgaa-mThiwW8nYAU\" 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=12850734&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cedbhgaa-mThiwW8nYAU&amp;vembed=0&amp;video_id=mThiwW8nYAU&amp;video_target=tpm-plugin-cedbhgaa-mThiwW8nYAU\" 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+5+-+System+of+Three+Equations+with+Three+Unknowns+Using+Elimination+(Infinite+Solutions)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 5: System of Three Equations with Three Unknowns Using Elimination (Infinite Solutions)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Applications<\/h2>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video example, we show how to define a system of three equations in three variables that represents a mixture needed by a chemist.\r\n<script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bfdghfec-612Ad0W9ZeY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/612Ad0W9ZeY?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bfdghfec-612Ad0W9ZeY\" 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=12850735&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bfdghfec-612Ad0W9ZeY&amp;vembed=0&amp;video_id=612Ad0W9ZeY&amp;video_target=tpm-plugin-bfdghfec-612Ad0W9ZeY\" 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\/System+of+3+Equations+with+3+Unknowns+Application+-+Concentration+Problem_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSystem of 3 Equations with 3 Unknowns Application - Concentration Problem\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Our last example shows you how to write a system of three equations that represents ticket sales for a theater that has three different prices for tickets.\r\n<script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-defacbch-Wg_v5R7BFo0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Wg_v5R7BFo0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-defacbch-Wg_v5R7BFo0\" 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=12850736&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-defacbch-Wg_v5R7BFo0&amp;vembed=0&amp;video_id=Wg_v5R7BFo0&amp;video_target=tpm-plugin-defacbch-Wg_v5R7BFo0\" 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\/System+of+3+Equations+with+3+Unknowns+Application+-+Ticket+Sales_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSystem of 3 Equations with 3 Unknowns Application - Ticket Sales\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>Solve systems of three equations in three variables.<\/li>\n<li>Identify inconsistent systems of equations containing three variables.<\/li>\n<li>Express the solution of a system of dependent equations containing three variables.<\/li>\n<\/ul>\n<\/section>\n<h2>Systems of Three Equations in Three Variables<\/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\">System Structure\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Each equation has form [latex]ax + by + cz = d[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Three equations with three unknowns ([latex]x[\/latex], [latex]y[\/latex], [latex]z[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Each equation represents a plane in 3D space<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Types of Solutions\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">One solution: [latex]{(x,y,z)}[\/latex] (intersection point of three planes)<\/li>\n<li class=\"whitespace-normal break-words\">No solution (inconsistent): leads to contradiction like [latex]3=0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Infinite solutions (dependent): leads to identity like [latex]0=0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2 aria-label=\"Try It\">Solve Systems of Three Equations in Three Variables<\/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\">Step-by-Step Solution Process\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Choose any two equations to work with first<\/li>\n<li class=\"whitespace-normal break-words\">Eliminate one variable to create a two-variable equation<\/li>\n<li class=\"whitespace-normal break-words\">Repeat with another pair to get a second two-variable equation<\/li>\n<li class=\"whitespace-normal break-words\">Solve the resulting two-by-two system<\/li>\n<li class=\"whitespace-normal break-words\">Use back-substitution to find the final variable<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Back-Substitution Strategy\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Once you find one variable, plug it into simpler equations<\/li>\n<li class=\"whitespace-normal break-words\">Work from simplest to most complex equations<\/li>\n<li class=\"whitespace-normal break-words\">Keep track of positive\/negative signs carefully<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Variable Selection\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Choose the variable that&#8217;s easiest to eliminate first<\/li>\n<li class=\"whitespace-normal break-words\">Look for equations where variables are missing<\/li>\n<li class=\"whitespace-normal break-words\">Consider coefficients that make elimination easier<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solution Verification\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Always check solution in ALL original equations<\/li>\n<li class=\"whitespace-normal break-words\">A true solution works in every equation<\/li>\n<li class=\"whitespace-normal break-words\">One equation being false means the solution is incorrect<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the following system of equations:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align*}  2x + y - 3z &= 4 \\  x - y + z &= 2 \\  3x + 2y &= 8  \\end{align*}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q428907\">Show Answer<\/button><\/p>\n<div id=\"q428907\" class=\"hidden-answer\" style=\"display: none\">\n[latex]x= \\frac{9}{4}, y= \\frac{5}{8}, z= \\frac{3}{8}[\/latex]\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video, we show another example of using back-substitution to solve a system in three variables.<br \/>\n<script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ededbedc-HHIjTChrIxE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/HHIjTChrIxE?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ededbedc-HHIjTChrIxE\" 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=12850729&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ededbedc-HHIjTChrIxE&amp;vembed=0&amp;video_id=HHIjTChrIxE&amp;video_target=tpm-plugin-ededbedc-HHIjTChrIxE\" 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+-+Solve+a+System+of+3+Equations+with+3+Unknowns+Using+Back+Substitution_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solve a System of 3 Equations with 3 Unknowns Using Back Substitution\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Solving a System of Three Equations in Three Variables by Elimination<\/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\">Preparation Steps\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Write all equations in standard form: [latex]ax + by + cz = d[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Clear any fractions by multiplying through<\/li>\n<li class=\"whitespace-normal break-words\">Label equations (1), (2), (3) for tracking<\/li>\n<li class=\"whitespace-normal break-words\">Plan which variable to eliminate first<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Elimination Process\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Choose same variable to eliminate from two pairs of equations<\/li>\n<li class=\"whitespace-normal break-words\">Create two new equations with two variables<\/li>\n<li class=\"whitespace-normal break-words\">Solve resulting two-by-two system<\/li>\n<li class=\"whitespace-normal break-words\">Back-substitute to find final variable<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Strategy\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Look for equations where coefficients are already opposites<\/li>\n<li class=\"whitespace-normal break-words\">Choose variable that&#8217;s easiest to eliminate<\/li>\n<li class=\"whitespace-normal break-words\">Keep coefficients as simple as possible<\/li>\n<li class=\"whitespace-normal break-words\">Create upper triangular form when possible<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Systematic Approach\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Always follow elimination steps in order<\/li>\n<li class=\"whitespace-normal break-words\">Track equations with numbers\/labels<\/li>\n<li class=\"whitespace-normal break-words\">Show all work clearly<\/li>\n<li class=\"whitespace-normal break-words\">Verify solution in all original equations<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the system of equations in three variables.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}2x+y - 2z=-1\\hfill \\\\ 3x - 3y-z=5\\hfill \\\\ x - 2y+3z=6\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q232738\">Show Solution<\/button><\/p>\n<div id=\"q232738\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(1,-1,1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following videos for more examples of the algebra you may encounter when solving systems with three variables.<br \/>\n<script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ehhecaff-r6htz3gaHZ0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/r6htz3gaHZ0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ehhecaff-r6htz3gaHZ0\" 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=12850730&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ehhecaff-r6htz3gaHZ0&amp;vembed=0&amp;video_id=r6htz3gaHZ0&amp;video_target=tpm-plugin-ehhecaff-r6htz3gaHZ0\" 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+-+System+of+Three+Equations+with+Three+Unknowns+Using+Elimination_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: System of Three Equations with Three Unknowns Using Elimination\u201d here (opens in new window).<\/a><\/p>\n<p><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-egffghab-3RbVSvvRyeI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/3RbVSvvRyeI?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-egffghab-3RbVSvvRyeI\" 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=12850731&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-egffghab-3RbVSvvRyeI&amp;vembed=0&amp;video_id=3RbVSvvRyeI&amp;video_target=tpm-plugin-egffghab-3RbVSvvRyeI\" 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+1+-+System+of+Three+Equations+with+Three+Unknowns+Using+Elimination_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: System of Three Equations with Three Unknowns Using Elimination\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video,\u00a0you will see a visual representation of the three possible outcomes for solutions to a system of equations in three variables. There is also a worked example of solving a system using elimination.<br \/>\n<script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-eabhhbcd-wIE8KSpb-E8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/wIE8KSpb-E8?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-eabhhbcd-wIE8KSpb-E8\" 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=12850732&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-eabhhbcd-wIE8KSpb-E8&amp;vembed=0&amp;video_id=wIE8KSpb-E8&amp;video_target=tpm-plugin-eabhhbcd-wIE8KSpb-E8\" 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\/Systems+of+Equations+in+Three+Variables+-+Part+1+of+2_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSystems of Equations in Three Variables: Part 1 of 2\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Inconsistent Systems of Equations Containing Three Variables<\/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\">Key Characteristics:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">No solution exists that satisfies all equations<\/li>\n<li class=\"whitespace-normal break-words\">Results in a contradiction (e.g., [latex]0 = 5[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Represents planes that don&#8217;t intersect at a common point<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Geometric Interpretations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Three parallel planes<\/li>\n<li class=\"whitespace-normal break-words\">Two parallel planes with one intersecting plane<\/li>\n<li class=\"whitespace-normal break-words\">Three planes intersecting in different locations<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Detection Method:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Elimination process leads to a contradiction<\/li>\n<li class=\"whitespace-normal break-words\">Often takes several steps to reveal the contradiction<\/li>\n<li class=\"whitespace-normal break-words\">Cannot be determined by looking at equations alone<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the system of three equations in three variables.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{ }x+y+z=2\\hfill \\\\ \\text{ }y - 3z=1\\hfill \\\\ 2x+y+5z=0\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q563392\">Show Solution<\/button><\/p>\n<div id=\"q563392\" class=\"hidden-answer\" style=\"display: none\">\n<p>No solution.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the video below for another example of using elimination in a system that has no solution.<br \/>\n<script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-aeghaacc-ryNQsWrUoJw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ryNQsWrUoJw?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-aeghaacc-ryNQsWrUoJw\" 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=12850733&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-aeghaacc-ryNQsWrUoJw&amp;vembed=0&amp;video_id=ryNQsWrUoJw&amp;video_target=tpm-plugin-aeghaacc-ryNQsWrUoJw\" 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+4+-+System+of+Three+Equations+with+Three+Unknowns+Using+Elimination+(No+Solution)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 4: System of Three Equations with Three Unknowns Using Elimination (No Solution)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Dependent Systems of Equations Containing Three Variables<\/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\">Key Characteristics:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Infinite solutions exist<\/li>\n<li class=\"whitespace-normal break-words\">Results in an identity (e.g., [latex]0 = 0[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Can express solution using one variable<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Geometric Interpretations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Three identical planes<\/li>\n<li class=\"whitespace-normal break-words\">Two identical planes intersecting a third<\/li>\n<li class=\"whitespace-normal break-words\">Three planes intersecting along a line<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the following system.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}x+y+z=7 \\\\ 3x - 2y-z=4 \\\\ x+6y+5z=24 \\end{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q195958\">Show Solution<\/button><\/p>\n<div id=\"q195958\" class=\"hidden-answer\" style=\"display: none\">\n<p>Infinitely many number of solutions of the form [latex]\\left(x,4x - 11,-5x+18\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">See the following video for another example of a dependent three-by-three system.<br \/>\n<script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cedbhgaa-mThiwW8nYAU\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/mThiwW8nYAU?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cedbhgaa-mThiwW8nYAU\" 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=12850734&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cedbhgaa-mThiwW8nYAU&amp;vembed=0&amp;video_id=mThiwW8nYAU&amp;video_target=tpm-plugin-cedbhgaa-mThiwW8nYAU\" 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+5+-+System+of+Three+Equations+with+Three+Unknowns+Using+Elimination+(Infinite+Solutions)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 5: System of Three Equations with Three Unknowns Using Elimination (Infinite Solutions)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Applications<\/h2>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video example, we show how to define a system of three equations in three variables that represents a mixture needed by a chemist.<br \/>\n<script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bfdghfec-612Ad0W9ZeY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/612Ad0W9ZeY?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bfdghfec-612Ad0W9ZeY\" 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=12850735&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bfdghfec-612Ad0W9ZeY&amp;vembed=0&amp;video_id=612Ad0W9ZeY&amp;video_target=tpm-plugin-bfdghfec-612Ad0W9ZeY\" 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\/System+of+3+Equations+with+3+Unknowns+Application+-+Concentration+Problem_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSystem of 3 Equations with 3 Unknowns Application &#8211; Concentration Problem\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Our last example shows you how to write a system of three equations that represents ticket sales for a theater that has three different prices for tickets.<br \/>\n<script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-defacbch-Wg_v5R7BFo0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Wg_v5R7BFo0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-defacbch-Wg_v5R7BFo0\" 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=12850736&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-defacbch-Wg_v5R7BFo0&amp;vembed=0&amp;video_id=Wg_v5R7BFo0&amp;video_target=tpm-plugin-defacbch-Wg_v5R7BFo0\" 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\/System+of+3+Equations+with+3+Unknowns+Application+-+Ticket+Sales_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSystem of 3 Equations with 3 Unknowns Application &#8211; Ticket Sales\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":19,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex: Solve a System of 3 Equations with 3 Unknowns Using Back Substitution\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/HHIjTChrIxE\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 2: System of Three Equations with Three Unknowns Using Elimination\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/r6htz3gaHZ0\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 1: System of Three Equations with Three Unknowns Using Elimination\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/3RbVSvvRyeI\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Systems of Equations in Three Variables: Part 1 of 2\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/wIE8KSpb-E8\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 4: System of Three Equations with Three Unknowns Using Elimination (No Solution)\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/ryNQsWrUoJw\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 5: System of Three Equations with Three Unknowns Using Elimination (Infinite Solutions)\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/mThiwW8nYAU\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"System of 3 Equations with 3 Unknowns Application - Concentration Problem\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/612Ad0W9ZeY\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"System of 3 Equations with 3 Unknowns Application - Ticket Sales\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/Wg_v5R7BFo0\",\"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":131,"module-header":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"Ex: Solve a System of 3 Equations with 3 Unknowns Using Back Substitution","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/HHIjTChrIxE","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Ex 2: System of Three Equations with Three Unknowns Using Elimination","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/r6htz3gaHZ0","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Ex 1: System of Three Equations with Three Unknowns Using Elimination","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/3RbVSvvRyeI","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Systems of Equations in Three Variables: Part 1 of 2","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/wIE8KSpb-E8","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Ex 4: System of Three Equations with Three Unknowns Using Elimination (No Solution)","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/ryNQsWrUoJw","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Ex 5: System of Three Equations with Three Unknowns Using Elimination (Infinite Solutions)","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/mThiwW8nYAU","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"System of 3 Equations with 3 Unknowns Application - Concentration Problem","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/612Ad0W9ZeY","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"System of 3 Equations with 3 Unknowns Application - Ticket Sales","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/Wg_v5R7BFo0","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=12850729&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ededbedc-HHIjTChrIxE&vembed=0&video_id=HHIjTChrIxE&video_target=tpm-plugin-ededbedc-HHIjTChrIxE'><\/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=12850730&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ehhecaff-r6htz3gaHZ0&vembed=0&video_id=r6htz3gaHZ0&video_target=tpm-plugin-ehhecaff-r6htz3gaHZ0'><\/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=12850731&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-egffghab-3RbVSvvRyeI&vembed=0&video_id=3RbVSvvRyeI&video_target=tpm-plugin-egffghab-3RbVSvvRyeI'><\/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=12850732&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-eabhhbcd-wIE8KSpb-E8&vembed=0&video_id=wIE8KSpb-E8&video_target=tpm-plugin-eabhhbcd-wIE8KSpb-E8'><\/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=12850733&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-aeghaacc-ryNQsWrUoJw&vembed=0&video_id=ryNQsWrUoJw&video_target=tpm-plugin-aeghaacc-ryNQsWrUoJw'><\/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=12850734&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-cedbhgaa-mThiwW8nYAU&vembed=0&video_id=mThiwW8nYAU&video_target=tpm-plugin-cedbhgaa-mThiwW8nYAU'><\/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=12850735&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bfdghfec-612Ad0W9ZeY&vembed=0&video_id=612Ad0W9ZeY&video_target=tpm-plugin-bfdghfec-612Ad0W9ZeY'><\/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=12850736&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-defacbch-Wg_v5R7BFo0&vembed=0&video_id=Wg_v5R7BFo0&video_target=tpm-plugin-defacbch-Wg_v5R7BFo0'><\/script>\n","media_targets":["tpm-plugin-ededbedc-HHIjTChrIxE","tpm-plugin-ehhecaff-r6htz3gaHZ0","tpm-plugin-egffghab-3RbVSvvRyeI","tpm-plugin-eabhhbcd-wIE8KSpb-E8","tpm-plugin-aeghaacc-ryNQsWrUoJw","tpm-plugin-cedbhgaa-mThiwW8nYAU","tpm-plugin-bfdghfec-612Ad0W9ZeY","tpm-plugin-defacbch-Wg_v5R7BFo0"]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1464"}],"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\/1464\/revisions"}],"predecessor-version":[{"id":3381,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1464\/revisions\/3381"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/131"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1464\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1464"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1464"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1464"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1464"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}