{"id":1454,"date":"2025-07-25T01:55:09","date_gmt":"2025-07-25T01:55:09","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1454"},"modified":"2026-03-24T07:27:57","modified_gmt":"2026-03-24T07:27:57","slug":"systems-of-linear-equations-two-variables-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/systems-of-linear-equations-two-variables-fresh-take\/","title":{"raw":"Systems of Linear Equations: Two Variables: Fresh Take","rendered":"Systems of Linear Equations: Two Variables: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Solve systems of equations by graphing.<\/li>\r\n \t<li>Solve systems of equations algebraically.<\/li>\r\n \t<li>Identify inconsistent and dependent systems of equations containing two variables.<\/li>\r\n \t<li>Solve applied systems.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Solutions of Systems Overview<\/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 Definition and Purpose:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A system contains two or more linear equations with two or more variables<\/li>\r\n \t<li class=\"whitespace-normal break-words\">All equations must be satisfied simultaneously for a solution<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solution requires finding specific values for all variables<\/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\">Independent systems: Exactly one solution (lines intersect at one point)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Inconsistent systems: No solution (parallel lines)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Dependent systems: Infinite solutions (same line)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solution Requirements:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Need at least as many equations as variables for a unique solution<\/li>\r\n \t<li class=\"whitespace-normal break-words\">This alone doesn't guarantee a unique solution<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solution must satisfy ALL equations in the system<\/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\">Substitute proposed solution into ALL equations<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Must get true statements for each equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graphically, solution point lies on all lines<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<ul>\r\n \t<li>A consistent system of equations has at least one solution.<\/li>\r\n \t<li>A consistent system is considered an <strong>independent system<\/strong> if it has a single solution.\r\n<ul>\r\n \t<li>Example: Two lines with different slopes intersect at one point in the plane.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>A consistent system is considered a <strong>dependent system<\/strong> if the equations have the same slope and the same y-intercepts.\r\n<ul>\r\n \t<li>The lines coincide, representing the same line.<\/li>\r\n \t<li>Every point on the line satisfies the system, so there are an infinite number of solutions.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>An <strong>inconsistent system<\/strong> is one in which the equations represent two parallel lines.\r\n<ul>\r\n \t<li>The lines have the same slope but different [latex]y[\/latex]-intercepts.<\/li>\r\n \t<li>There are no points common to both lines, so there is no solution to the system.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Determine whether the ordered pair [latex]\\left(8,5\\right)[\/latex] is a solution to the following system.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}5x - 4y=20\\\\ 2x+1=3y\\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"672974\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"672974\"]Not a solution.[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video for another example of how to verify whether an ordered pair is a solution to a system of equations.\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-dfhfgghf-2IxgKgjX00k\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/2IxgKgjX00k?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dfhfgghf-2IxgKgjX00k\" 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=12850479&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dfhfgghf-2IxgKgjX00k&amp;vembed=0&amp;video_id=2IxgKgjX00k&amp;video_target=tpm-plugin-dfhfgghf-2IxgKgjX00k\" 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\/Determine+if+an+Ordered+Pair+is+a+Solution+to+a+System+of+Linear+Equations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermine if an Ordered Pair is a Solution to a System of Linear Equations\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Solving Systems of Equations by Graphing<\/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\">Graphical Approach Steps:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Graph both equations on the same coordinate plane<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find the point of intersection (if it exists)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Verify the solution algebraically<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identify the system type based on how lines intersect<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">System Types from Graphs:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Intersecting lines \u2192 One solution (independent)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Parallel lines \u2192 No solution (inconsistent)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Same line \u2192 Infinite solutions (dependent)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graphing Techniques:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Solve for [latex]y[\/latex] to get slope-intercept form<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use slope and [latex]y[\/latex]-intercept method<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use [latex]x[\/latex] and [latex]y[\/latex] intercepts method<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Plot strategic points<\/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 intersection point in both equations<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Both equations must be satisfied<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graphical solutions are approximate unless points are clearly identifiable<\/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 by graphing.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}2x - 5y=-25 \\\\ -4x+5y=35 \\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"141689\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"141689\"]\r\n\r\nThe solution to the system is the ordered pair [latex]\\left(-5,3\\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video for an example of how to identify what type of system and solution are represented in the graph of a 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-hbgfgfcc-ZolxtOjcEQY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ZolxtOjcEQY?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hbgfgfcc-ZolxtOjcEQY\" 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=12850648&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hbgfgfcc-ZolxtOjcEQY&amp;vembed=0&amp;video_id=ZolxtOjcEQY&amp;video_target=tpm-plugin-hbgfgfcc-ZolxtOjcEQY\" 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\/Determine+the+Number+of+Solutions+to+a+System+of+Linear+Equations+From+a+Graph_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermine the Number of Solutions to a System of Linear Equations From a Graph\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the next video to see how to solve a system of equations by first graphing the lines and then identifying the type of 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-gceeched-Lv832rXAQ5k\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Lv832rXAQ5k?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gceeched-Lv832rXAQ5k\" 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=12850649&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gceeched-Lv832rXAQ5k&amp;vembed=0&amp;video_id=Lv832rXAQ5k&amp;video_target=tpm-plugin-gceeched-Lv832rXAQ5k\" 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+Equations+by+Graphing_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Solve a System of Equations by Graphing\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Solving Systems of Equations by Substitution<\/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\">Method Overview:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Solve one equation for one variable<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute that expression into other equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve resulting equation for remaining variable<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Back-substitute to find first variable<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Advantages:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">More precise than graphing<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Works well with fractional\/decimal solutions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Often simpler than other algebraic methods<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solution Analysis:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">May reveal no solution (contradictory equations)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">May reveal infinite solutions (identical equations)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check final answer in both 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 using substitution:<center>[latex]\\begin{align}\r\ny = 3x + 1 \\\\\r\n2x - y = 5\r\n\\end{align}[\/latex]<\/center>[reveal-answer q=\"238943\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"238943\"]\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">First equation is already solved for y, so substitute [latex]y = 3x + 1[\/latex] into second equation:\r\n<center>[latex]2x - (3x + 1) = 5[\/latex]<\/center><\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve for [latex]x[\/latex]:\r\n<center>[latex]\r\n\\begin{array}{rcl}\r\n2x - 3x - 1 = 5 \\\\\r\n-x - 1 = 5 \\\\\r\n-x = 6 \\\\\r\nx = -6\r\n\\end{array}\r\n[\/latex]<\/center><\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute [latex]x = -6[\/latex] back into [latex]y = 3x + 1[\/latex]:\r\n<center>[latex]\r\n\\begin{array}{rcl}\r\ny = 3(-6) + 1 \\\\\r\ny = -18 + 1 \\\\\r\ny = -17\r\n\\end{array}\r\n[\/latex]<\/center><\/li>\r\n \t<li class=\"whitespace-normal break-words\">Verify solution [latex](-6, -17)[\/latex] in both equations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Equation 1:<center>[latex]-17 \\stackrel{?}{=} 3(-6) + 1[\/latex]<\/center>\r\n<center>[latex]-17 = -18 + 1 = -17[\/latex] \u2713<\/center><\/li>\r\n \t<li class=\"whitespace-normal break-words\">Equation 2:<center>[latex]2(-6) - (-17) \\stackrel{?}{=} 5[\/latex]<\/center>\r\n<center>[latex]-12 + 17 = 5[\/latex] \u2713<\/center><\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the solution is [latex](-6, -17)[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video, you will be given an example of solving a system of two equations using the substitution method.\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-ghaagdhc-MIXL35YRzRw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/MIXL35YRzRw?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ghaagdhc-MIXL35YRzRw\" 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=12850650&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ghaagdhc-MIXL35YRzRw&amp;vembed=0&amp;video_id=MIXL35YRzRw&amp;video_target=tpm-plugin-ghaagdhc-MIXL35YRzRw\" 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+Equations+Using+Substitution_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Solve a System of Equations Using Substitution\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">The following video is ~10 minutes long and provides a mini-lesson on using the substitution method to solve a system of linear equations. \u00a0We present three different examples, and also use a graphing tool to help summarize the solution for each example.\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-gcbbbcgg-HxhacvH49o8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/HxhacvH49o8?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gcbbbcgg-HxhacvH49o8\" 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=12850651&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gcbbbcgg-HxhacvH49o8&amp;vembed=0&amp;video_id=HxhacvH49o8&amp;video_target=tpm-plugin-gcbbbcgg-HxhacvH49o8\" 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\/Solving+Systems+of+Equations+using+Substitution_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolving Systems of Equations using Substitution\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Solving Systems of Equations by the Addition Method<\/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\">Method Fundamentals:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Also called elimination method<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Add equations to eliminate one variable<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Variables with opposite coefficients sum to zero<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Goal is to isolate one variable<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Steps:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Align like terms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiply equations if needed for elimination<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Add equations to eliminate one variable<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve for remaining variable<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Back-substitute to find other variable<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Strategic Planning:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Look for variables with opposite coefficients<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find LCM of coefficients if multiplication needed<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Choose simpler variable to eliminate first<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Clear fractions before adding if needed<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">When to Use:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Variables have coefficients that are opposites<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Coefficients have a small LCM<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitution would be too complex<\/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 by addition.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}2x - 7y&amp;=2\\\\ 3x+y&amp;=-20\\end{align}[\/latex]<\/p>\r\n[reveal-answer q=\"609174\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"609174\"]\r\n\r\n[latex]\\left(-6,-2\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve the system of equations by addition.[latex]\\begin{align}2x+3y&amp;=8\\\\ 3x+5y&amp;=10\\end{align}[\/latex][reveal-answer q=\"326265\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"326265\"][latex]\\left(10,-4\\right)[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video, you will see another example of how to use the method of elimination to solve a system of linear equations.\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-beccaeec-M4IEmwcqR3c\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/M4IEmwcqR3c?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-beccaeec-M4IEmwcqR3c\" 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=12850652&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-beccaeec-M4IEmwcqR3c&amp;vembed=0&amp;video_id=M4IEmwcqR3c&amp;video_target=tpm-plugin-beccaeec-M4IEmwcqR3c\" 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+-+Solve+a+System+of+Equations+Using+the+Elimination+Method_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Solve a System of Equations Using the Elimination Method\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Below is another video example of using the elimination method to solve a system of linear equations when multiplication of one equation is required.\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-chfghbfc-_liDhKops2w\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/_liDhKops2w?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-chfghbfc-_liDhKops2w\" 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=12850653&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-chfghbfc-_liDhKops2w&amp;vembed=0&amp;video_id=_liDhKops2w&amp;video_target=tpm-plugin-chfghbfc-_liDhKops2w\" 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+Equations+Using+the+Elimination+Method_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Solve a System of Equations Using the Elimination Method\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video, you will find one more example of using the elimination method to solve a system; this one has coefficients that are fractions.\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-effcdhfd-s3S64b1DrtQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/s3S64b1DrtQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-effcdhfd-s3S64b1DrtQ\" 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=12850654&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-effcdhfd-s3S64b1DrtQ&amp;vembed=0&amp;video_id=s3S64b1DrtQ&amp;video_target=tpm-plugin-effcdhfd-s3S64b1DrtQ\" 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+Equations+Using+Eliminations+(Fractions)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solve a System of Equations Using Eliminations (Fractions)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video we present more examples of how to use the addition (elimination) method to solve a system of two linear equations.\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-cecbggea-ova8GSmPV4o\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ova8GSmPV4o?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cecbggea-ova8GSmPV4o\" 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=12850655&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cecbggea-ova8GSmPV4o&amp;vembed=0&amp;video_id=ova8GSmPV4o&amp;video_target=tpm-plugin-cecbggea-ova8GSmPV4o\" 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\/Solving+Systems+of+Equations+using+Elimination_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolving Systems of Equations using Elimination\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Identifying Inconsistent Systems of Equations Containing Two 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\">Understanding Inconsistent Systems:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Parallel lines that never intersect<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Same slope but different [latex]y[\/latex]-intercepts<\/li>\r\n \t<li class=\"whitespace-normal break-words\">No solution exists<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Result in contradictory equations<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Recognition Methods:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Converting to slope-intercept form reveals parallel lines<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solving leads to contradictory statement (e.g., [latex]5 = 8[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Algebraic solution gives impossible result<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graphing shows parallel lines<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Verification Techniques:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Write equations in slope-intercept form<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Compare slopes and [latex]y[\/latex]-intercepts<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Attempt algebraic solution<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graph equations to confirm parallel lines<\/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 in two variables.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}2y - 2x=2\\\\ 2y - 2x=6\\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"681787\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"681787\"]\r\n\r\nNo solution. It is an inconsistent system.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">In the next video, see another example of using substitution to solve 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-ahddabhe-kTtKfh5gFUc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/kTtKfh5gFUc?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ahddabhe-kTtKfh5gFUc\" 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=12850656&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ahddabhe-kTtKfh5gFUc&amp;vembed=0&amp;video_id=kTtKfh5gFUc&amp;video_target=tpm-plugin-ahddabhe-kTtKfh5gFUc\" 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+Equations+Using+Substitution+-+No+Solution_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solve a System of Equations Using Substitution - No Solution\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Expressing the Solution of a System of Dependent Equations Containing Two 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\">Understanding Dependent Systems:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Two equations represent the same line<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Infinite number of solutions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">All points on one line satisfy both equations<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solving yields identity (e.g., [latex]0 = 0[\/latex])<\/li>\r\n<\/ul>\r\n<\/li>\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\">Same slope and same [latex]y[\/latex]-intercept<\/li>\r\n \t<li class=\"whitespace-normal break-words\">One equation is multiple of the other<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Equations reduce to identity<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Lines are coincident (overlap completely)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solution Format:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Written in parametric form: [latex](x, mx + b)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]x[\/latex] can be any real number<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y[\/latex] is expressed in terms of [latex]x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Represents all points on the line<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identification Methods:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Convert to slope-intercept form<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Compare coefficients after simplification<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Look for scalar multiples<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check if equations are equivalent<\/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 in two variables.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}y - 2x=5 \\\\ -3y+6x=-15 \\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"218404\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"218404\"]\r\n\r\nThe system is dependent so there are infinitely many solutions of the form [latex]\\left(x,2x+5\\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the next video for an example of a dependent 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-acghhhah-Pcqb109yK5Q\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Pcqb109yK5Q?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-acghhhah-Pcqb109yK5Q\" 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=12850657&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-acghhhah-Pcqb109yK5Q&amp;vembed=0&amp;video_id=Pcqb109yK5Q&amp;video_target=tpm-plugin-acghhhah-Pcqb109yK5Q\" 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+Equations+Using+Substitution+-+Infinite+Solutions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solve a System of Equations Using Substitution - Infinite Solutions\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Application Problems<\/h2>\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-ccfgcadf-qey3FmE8saQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/qey3FmE8saQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ccfgcadf-qey3FmE8saQ\" 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=12850727&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ccfgcadf-qey3FmE8saQ&amp;vembed=0&amp;video_id=qey3FmE8saQ&amp;video_target=tpm-plugin-ccfgcadf-qey3FmE8saQ\" 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+Equations+App+-++Break-Even+Point_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSystem of Equations App: Break-Even Point\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-gadedcah-euh9ksWrq0A\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/euh9ksWrq0A?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gadedcah-euh9ksWrq0A\" 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=12850728&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gadedcah-euh9ksWrq0A&amp;vembed=0&amp;video_id=euh9ksWrq0A&amp;video_target=tpm-plugin-gadedcah-euh9ksWrq0A\" 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+an+Application+Problem+Using+a+System+of+Linear+Equations+(09x-43)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solve an Application Problem Using a System of Linear Equations (09x-43)\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 equations by graphing.<\/li>\n<li>Solve systems of equations algebraically.<\/li>\n<li>Identify inconsistent and dependent systems of equations containing two variables.<\/li>\n<li>Solve applied systems.<\/li>\n<\/ul>\n<\/section>\n<h2>Solutions of Systems Overview<\/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 Definition and Purpose:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A system contains two or more linear equations with two or more variables<\/li>\n<li class=\"whitespace-normal break-words\">All equations must be satisfied simultaneously for a solution<\/li>\n<li class=\"whitespace-normal break-words\">Solution requires finding specific values for all variables<\/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\">Independent systems: Exactly one solution (lines intersect at one point)<\/li>\n<li class=\"whitespace-normal break-words\">Inconsistent systems: No solution (parallel lines)<\/li>\n<li class=\"whitespace-normal break-words\">Dependent systems: Infinite solutions (same line)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solution Requirements:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Need at least as many equations as variables for a unique solution<\/li>\n<li class=\"whitespace-normal break-words\">This alone doesn&#8217;t guarantee a unique solution<\/li>\n<li class=\"whitespace-normal break-words\">Solution must satisfy ALL equations in the system<\/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\">Substitute proposed solution into ALL equations<\/li>\n<li class=\"whitespace-normal break-words\">Must get true statements for each equation<\/li>\n<li class=\"whitespace-normal break-words\">Graphically, solution point lies on all lines<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<ul>\n<li>A consistent system of equations has at least one solution.<\/li>\n<li>A consistent system is considered an <strong>independent system<\/strong> if it has a single solution.\n<ul>\n<li>Example: Two lines with different slopes intersect at one point in the plane.<\/li>\n<\/ul>\n<\/li>\n<li>A consistent system is considered a <strong>dependent system<\/strong> if the equations have the same slope and the same y-intercepts.\n<ul>\n<li>The lines coincide, representing the same line.<\/li>\n<li>Every point on the line satisfies the system, so there are an infinite number of solutions.<\/li>\n<\/ul>\n<\/li>\n<li>An <strong>inconsistent system<\/strong> is one in which the equations represent two parallel lines.\n<ul>\n<li>The lines have the same slope but different [latex]y[\/latex]-intercepts.<\/li>\n<li>There are no points common to both lines, so there is no solution to the system.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Determine whether the ordered pair [latex]\\left(8,5\\right)[\/latex] is a solution to the following system.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}5x - 4y=20\\\\ 2x+1=3y\\end{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q672974\">Show Solution<\/button><\/p>\n<div id=\"q672974\" class=\"hidden-answer\" style=\"display: none\">Not a solution.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video for another example of how to verify whether an ordered pair is a solution to a system of equations.<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-dfhfgghf-2IxgKgjX00k\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/2IxgKgjX00k?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dfhfgghf-2IxgKgjX00k\" 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=12850479&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dfhfgghf-2IxgKgjX00k&amp;vembed=0&amp;video_id=2IxgKgjX00k&amp;video_target=tpm-plugin-dfhfgghf-2IxgKgjX00k\" 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\/Determine+if+an+Ordered+Pair+is+a+Solution+to+a+System+of+Linear+Equations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermine if an Ordered Pair is a Solution to a System of Linear Equations\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Solving Systems of Equations by Graphing<\/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\">Graphical Approach Steps:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Graph both equations on the same coordinate plane<\/li>\n<li class=\"whitespace-normal break-words\">Find the point of intersection (if it exists)<\/li>\n<li class=\"whitespace-normal break-words\">Verify the solution algebraically<\/li>\n<li class=\"whitespace-normal break-words\">Identify the system type based on how lines intersect<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">System Types from Graphs:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Intersecting lines \u2192 One solution (independent)<\/li>\n<li class=\"whitespace-normal break-words\">Parallel lines \u2192 No solution (inconsistent)<\/li>\n<li class=\"whitespace-normal break-words\">Same line \u2192 Infinite solutions (dependent)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Graphing Techniques:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Solve for [latex]y[\/latex] to get slope-intercept form<\/li>\n<li class=\"whitespace-normal break-words\">Use slope and [latex]y[\/latex]-intercept method<\/li>\n<li class=\"whitespace-normal break-words\">Use [latex]x[\/latex] and [latex]y[\/latex] intercepts method<\/li>\n<li class=\"whitespace-normal break-words\">Plot strategic points<\/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 intersection point in both equations<\/li>\n<li class=\"whitespace-normal break-words\">Both equations must be satisfied<\/li>\n<li class=\"whitespace-normal break-words\">Graphical solutions are approximate unless points are clearly identifiable<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the following system of equations by graphing.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}2x - 5y=-25 \\\\ -4x+5y=35 \\end{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q141689\">Show Solution<\/button><\/p>\n<div id=\"q141689\" class=\"hidden-answer\" style=\"display: none\">\n<p>The solution to the system is the ordered pair [latex]\\left(-5,3\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video for an example of how to identify what type of system and solution are represented in the graph of a 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-hbgfgfcc-ZolxtOjcEQY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ZolxtOjcEQY?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hbgfgfcc-ZolxtOjcEQY\" 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=12850648&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hbgfgfcc-ZolxtOjcEQY&amp;vembed=0&amp;video_id=ZolxtOjcEQY&amp;video_target=tpm-plugin-hbgfgfcc-ZolxtOjcEQY\" 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\/Determine+the+Number+of+Solutions+to+a+System+of+Linear+Equations+From+a+Graph_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermine the Number of Solutions to a System of Linear Equations From a Graph\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the next video to see how to solve a system of equations by first graphing the lines and then identifying the type of 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-gceeched-Lv832rXAQ5k\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Lv832rXAQ5k?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gceeched-Lv832rXAQ5k\" 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=12850649&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gceeched-Lv832rXAQ5k&amp;vembed=0&amp;video_id=Lv832rXAQ5k&amp;video_target=tpm-plugin-gceeched-Lv832rXAQ5k\" 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+Equations+by+Graphing_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Solve a System of Equations by Graphing\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Solving Systems of Equations by Substitution<\/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\">Method Overview:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Solve one equation for one variable<\/li>\n<li class=\"whitespace-normal break-words\">Substitute that expression into other equation<\/li>\n<li class=\"whitespace-normal break-words\">Solve resulting equation for remaining variable<\/li>\n<li class=\"whitespace-normal break-words\">Back-substitute to find first variable<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Advantages:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">More precise than graphing<\/li>\n<li class=\"whitespace-normal break-words\">Works well with fractional\/decimal solutions<\/li>\n<li class=\"whitespace-normal break-words\">Often simpler than other algebraic methods<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solution Analysis:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">May reveal no solution (contradictory equations)<\/li>\n<li class=\"whitespace-normal break-words\">May reveal infinite solutions (identical equations)<\/li>\n<li class=\"whitespace-normal break-words\">Check final answer in both original equations<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the system of equations using substitution:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}  y = 3x + 1 \\\\  2x - y = 5  \\end{align}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q238943\">Show Answer<\/button><\/p>\n<div id=\"q238943\" class=\"hidden-answer\" style=\"display: none\">\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">First equation is already solved for y, so substitute [latex]y = 3x + 1[\/latex] into second equation:\n<div style=\"text-align: center;\">[latex]2x - (3x + 1) = 5[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solve for [latex]x[\/latex]:\n<div style=\"text-align: center;\">[latex]\\begin{array}{rcl}  2x - 3x - 1 = 5 \\\\  -x - 1 = 5 \\\\  -x = 6 \\\\  x = -6  \\end{array}[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">Substitute [latex]x = -6[\/latex] back into [latex]y = 3x + 1[\/latex]:\n<div style=\"text-align: center;\">[latex]\\begin{array}{rcl}  y = 3(-6) + 1 \\\\  y = -18 + 1 \\\\  y = -17  \\end{array}[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">Verify solution [latex](-6, -17)[\/latex] in both equations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Equation 1:\n<div style=\"text-align: center;\">[latex]-17 \\stackrel{?}{=} 3(-6) + 1[\/latex]<\/div>\n<div style=\"text-align: center;\">[latex]-17 = -18 + 1 = -17[\/latex] \u2713<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">Equation 2:\n<div style=\"text-align: center;\">[latex]2(-6) - (-17) \\stackrel{?}{=} 5[\/latex]<\/div>\n<div style=\"text-align: center;\">[latex]-12 + 17 = 5[\/latex] \u2713<\/div>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the solution is [latex](-6, -17)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video, you will be given an example of solving a system of two equations using the substitution method.<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-ghaagdhc-MIXL35YRzRw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/MIXL35YRzRw?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ghaagdhc-MIXL35YRzRw\" 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=12850650&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ghaagdhc-MIXL35YRzRw&amp;vembed=0&amp;video_id=MIXL35YRzRw&amp;video_target=tpm-plugin-ghaagdhc-MIXL35YRzRw\" 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+Equations+Using+Substitution_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Solve a System of Equations Using Substitution\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">The following video is ~10 minutes long and provides a mini-lesson on using the substitution method to solve a system of linear equations. \u00a0We present three different examples, and also use a graphing tool to help summarize the solution for each example.<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-gcbbbcgg-HxhacvH49o8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/HxhacvH49o8?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gcbbbcgg-HxhacvH49o8\" 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=12850651&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gcbbbcgg-HxhacvH49o8&amp;vembed=0&amp;video_id=HxhacvH49o8&amp;video_target=tpm-plugin-gcbbbcgg-HxhacvH49o8\" 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\/Solving+Systems+of+Equations+using+Substitution_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolving Systems of Equations using Substitution\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Solving Systems of Equations by the Addition Method<\/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\">Method Fundamentals:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Also called elimination method<\/li>\n<li class=\"whitespace-normal break-words\">Add equations to eliminate one variable<\/li>\n<li class=\"whitespace-normal break-words\">Variables with opposite coefficients sum to zero<\/li>\n<li class=\"whitespace-normal break-words\">Goal is to isolate one variable<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Steps:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Align like terms<\/li>\n<li class=\"whitespace-normal break-words\">Multiply equations if needed for elimination<\/li>\n<li class=\"whitespace-normal break-words\">Add equations to eliminate one variable<\/li>\n<li class=\"whitespace-normal break-words\">Solve for remaining variable<\/li>\n<li class=\"whitespace-normal break-words\">Back-substitute to find other variable<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Strategic Planning:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Look for variables with opposite coefficients<\/li>\n<li class=\"whitespace-normal break-words\">Find LCM of coefficients if multiplication needed<\/li>\n<li class=\"whitespace-normal break-words\">Choose simpler variable to eliminate first<\/li>\n<li class=\"whitespace-normal break-words\">Clear fractions before adding if needed<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">When to Use:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Variables have coefficients that are opposites<\/li>\n<li class=\"whitespace-normal break-words\">Coefficients have a small LCM<\/li>\n<li class=\"whitespace-normal break-words\">Substitution would be too complex<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the system of equations by addition.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}2x - 7y&=2\\\\ 3x+y&=-20\\end{align}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q609174\">Show Solution<\/button><\/p>\n<div id=\"q609174\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(-6,-2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the system of equations by addition.[latex]\\begin{align}2x+3y&=8\\\\ 3x+5y&=10\\end{align}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q326265\">Show Solution<\/button><\/p>\n<div id=\"q326265\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left(10,-4\\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video, you will see another example of how to use the method of elimination to solve a system of linear equations.<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-beccaeec-M4IEmwcqR3c\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/M4IEmwcqR3c?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-beccaeec-M4IEmwcqR3c\" 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=12850652&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-beccaeec-M4IEmwcqR3c&amp;vembed=0&amp;video_id=M4IEmwcqR3c&amp;video_target=tpm-plugin-beccaeec-M4IEmwcqR3c\" 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+-+Solve+a+System+of+Equations+Using+the+Elimination+Method_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Solve a System of Equations Using the Elimination Method\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Below is another video example of using the elimination method to solve a system of linear equations when multiplication of one equation is required.<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-chfghbfc-_liDhKops2w\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/_liDhKops2w?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-chfghbfc-_liDhKops2w\" 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=12850653&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-chfghbfc-_liDhKops2w&amp;vembed=0&amp;video_id=_liDhKops2w&amp;video_target=tpm-plugin-chfghbfc-_liDhKops2w\" 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+Equations+Using+the+Elimination+Method_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Solve a System of Equations Using the Elimination Method\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video, you will find one more example of using the elimination method to solve a system; this one has coefficients that are fractions.<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-effcdhfd-s3S64b1DrtQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/s3S64b1DrtQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-effcdhfd-s3S64b1DrtQ\" 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=12850654&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-effcdhfd-s3S64b1DrtQ&amp;vembed=0&amp;video_id=s3S64b1DrtQ&amp;video_target=tpm-plugin-effcdhfd-s3S64b1DrtQ\" 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+Equations+Using+Eliminations+(Fractions)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solve a System of Equations Using Eliminations (Fractions)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video we present more examples of how to use the addition (elimination) method to solve a system of two linear equations.<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-cecbggea-ova8GSmPV4o\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ova8GSmPV4o?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cecbggea-ova8GSmPV4o\" 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=12850655&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cecbggea-ova8GSmPV4o&amp;vembed=0&amp;video_id=ova8GSmPV4o&amp;video_target=tpm-plugin-cecbggea-ova8GSmPV4o\" 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\/Solving+Systems+of+Equations+using+Elimination_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolving Systems of Equations using Elimination\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2 data-type=\"title\">Identifying Inconsistent Systems of Equations Containing Two 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\">Understanding Inconsistent Systems:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Parallel lines that never intersect<\/li>\n<li class=\"whitespace-normal break-words\">Same slope but different [latex]y[\/latex]-intercepts<\/li>\n<li class=\"whitespace-normal break-words\">No solution exists<\/li>\n<li class=\"whitespace-normal break-words\">Result in contradictory equations<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Recognition Methods:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Converting to slope-intercept form reveals parallel lines<\/li>\n<li class=\"whitespace-normal break-words\">Solving leads to contradictory statement (e.g., [latex]5 = 8[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Algebraic solution gives impossible result<\/li>\n<li class=\"whitespace-normal break-words\">Graphing shows parallel lines<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Verification Techniques:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Write equations in slope-intercept form<\/li>\n<li class=\"whitespace-normal break-words\">Compare slopes and [latex]y[\/latex]-intercepts<\/li>\n<li class=\"whitespace-normal break-words\">Attempt algebraic solution<\/li>\n<li class=\"whitespace-normal break-words\">Graph equations to confirm parallel lines<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the following system of equations in two variables.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}2y - 2x=2\\\\ 2y - 2x=6\\end{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q681787\">Show Solution<\/button><\/p>\n<div id=\"q681787\" class=\"hidden-answer\" style=\"display: none\">\n<p>No solution. It is an inconsistent system.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">In the next video, see another example of using substitution to solve 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-ahddabhe-kTtKfh5gFUc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/kTtKfh5gFUc?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ahddabhe-kTtKfh5gFUc\" 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=12850656&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ahddabhe-kTtKfh5gFUc&amp;vembed=0&amp;video_id=kTtKfh5gFUc&amp;video_target=tpm-plugin-ahddabhe-kTtKfh5gFUc\" 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+Equations+Using+Substitution+-+No+Solution_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solve a System of Equations Using Substitution &#8211; No Solution\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Expressing the Solution of a System of Dependent Equations Containing Two 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\">Understanding Dependent Systems:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Two equations represent the same line<\/li>\n<li class=\"whitespace-normal break-words\">Infinite number of solutions<\/li>\n<li class=\"whitespace-normal break-words\">All points on one line satisfy both equations<\/li>\n<li class=\"whitespace-normal break-words\">Solving yields identity (e.g., [latex]0 = 0[\/latex])<\/li>\n<\/ul>\n<\/li>\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\">Same slope and same [latex]y[\/latex]-intercept<\/li>\n<li class=\"whitespace-normal break-words\">One equation is multiple of the other<\/li>\n<li class=\"whitespace-normal break-words\">Equations reduce to identity<\/li>\n<li class=\"whitespace-normal break-words\">Lines are coincident (overlap completely)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solution Format:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Written in parametric form: [latex](x, mx + b)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x[\/latex] can be any real number<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y[\/latex] is expressed in terms of [latex]x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Represents all points on the line<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Identification Methods:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Convert to slope-intercept form<\/li>\n<li class=\"whitespace-normal break-words\">Compare coefficients after simplification<\/li>\n<li class=\"whitespace-normal break-words\">Look for scalar multiples<\/li>\n<li class=\"whitespace-normal break-words\">Check if equations are equivalent<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the following system of equations in two variables.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}y - 2x=5 \\\\ -3y+6x=-15 \\end{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q218404\">Show Solution<\/button><\/p>\n<div id=\"q218404\" class=\"hidden-answer\" style=\"display: none\">\n<p>The system is dependent so there are infinitely many solutions of the form [latex]\\left(x,2x+5\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the next video for an example of a dependent 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-acghhhah-Pcqb109yK5Q\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Pcqb109yK5Q?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-acghhhah-Pcqb109yK5Q\" 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=12850657&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-acghhhah-Pcqb109yK5Q&amp;vembed=0&amp;video_id=Pcqb109yK5Q&amp;video_target=tpm-plugin-acghhhah-Pcqb109yK5Q\" 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+Equations+Using+Substitution+-+Infinite+Solutions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solve a System of Equations Using Substitution &#8211; Infinite Solutions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Application Problems<\/h2>\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-ccfgcadf-qey3FmE8saQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/qey3FmE8saQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ccfgcadf-qey3FmE8saQ\" 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=12850727&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ccfgcadf-qey3FmE8saQ&amp;vembed=0&amp;video_id=qey3FmE8saQ&amp;video_target=tpm-plugin-ccfgcadf-qey3FmE8saQ\" 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+Equations+App+-++Break-Even+Point_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSystem of Equations App: Break-Even Point\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-gadedcah-euh9ksWrq0A\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/euh9ksWrq0A?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gadedcah-euh9ksWrq0A\" 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=12850728&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gadedcah-euh9ksWrq0A&amp;vembed=0&amp;video_id=euh9ksWrq0A&amp;video_target=tpm-plugin-gadedcah-euh9ksWrq0A\" 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+an+Application+Problem+Using+a+System+of+Linear+Equations+(09x-43)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solve an Application Problem Using a System of Linear Equations (09x-43)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Determine if an Ordered Pair is a Solution to a System of Linear Equations\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/2IxgKgjX00k\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Determine the Number of Solutions to a System of Linear Equations From a Graph\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/ZolxtOjcEQY\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard Youtube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 2: Solve a System of Equations by Graphing\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/Lv832rXAQ5k\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 2: Solve a System of Equations Using Substitution\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/MIXL35YRzRw\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Solving Systems of Equations using Substitution\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/HxhacvH49o8\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 1: Solve a System of Equations Using the Elimination Method\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/M4IEmwcqR3c\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 2: Solve a System of Equations Using the Elimination Method\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/_liDhKops2w\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Solve a System of Equations Using Eliminations (Fractions)\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/s3S64b1DrtQ\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Solving Systems of Equations using Elimination\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/ova8GSmPV4o\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Solve a System of Equations Using Substitution - No Solution\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/kTtKfh5gFUc\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Solve a System of Equations Using Substitution - Infinite Solutions\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/Pcqb109yK5Q\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"System of Equations App: Break-Even Point\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/qey3FmE8saQ\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Solve an Application Problem Using a System of Linear Equations (09x-43)\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/euh9ksWrq0A\",\"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":"Determine if an Ordered Pair is a Solution to a System of Linear Equations","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/2IxgKgjX00k","project":"","license":"arr","license_terms":"Standard YouTube 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