{"id":1457,"date":"2025-07-25T01:56:01","date_gmt":"2025-07-25T01:56:01","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1457"},"modified":"2026-03-24T07:22:37","modified_gmt":"2026-03-24T07:22:37","slug":"systems-of-nonlinear-equations-and-inequalities-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/systems-of-nonlinear-equations-and-inequalities-fresh-take\/","title":{"raw":"Systems of Nonlinear Equations and Inequalities: Fresh Take","rendered":"Systems of Nonlinear Equations and Inequalities: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Solve a system of nonlinear equations.<\/li>\r\n \t<li>Graph a system of nonlinear inequalities.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Solving a System of Nonlinear Equations Using Substitution<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p class=\"font-600 text-lg font-bold\"><strong>Common Features<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Solution Types\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">No solutions (no intersection)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">One solution (tangent point)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Two solutions (line crosses through curve)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solution Method\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Always start with linear equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve linear equation for one variable<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute into nonlinear equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve resulting equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check all solutions in both equations<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-lg font-bold\"><strong>Line-Parabola Systems<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Form\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Line: [latex]ax + by = c[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Parabola: [latex]y = ax^2 + bx + c[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solution Process\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Results in quadratic equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Two solutions: Line crosses parabola<\/li>\r\n \t<li class=\"whitespace-normal break-words\">One solution: Line tangent to parabola<\/li>\r\n \t<li class=\"whitespace-normal break-words\">No solutions: Line misses parabola<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-lg font-bold\"><strong>Line-Circle Systems<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Form\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Line: [latex]ax + by = c[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Circle: [latex](x - h)^2 + (y - k)^2 = r^2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solution Process\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Results in quadratic equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Two solutions: Line crosses circle<\/li>\r\n \t<li class=\"whitespace-normal break-words\">One solution: Line tangent to circle<\/li>\r\n \t<li class=\"whitespace-normal break-words\">No solutions: Line outside circle<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Solve the given system of equations by substitution.\r\n<div style=\"text-align: center;\">[latex]\\begin{gathered}3x-y=-2 \\\\ 2{x}^{2}-y=0 \\end{gathered}[\/latex]<\/div>\r\n&nbsp;\r\n<p style=\"text-align: left;\">[reveal-answer q=\"503330\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"503330\"]<\/p>\r\n[latex]\\left(-\\dfrac{1}{2},\\dfrac{1}{2}\\right)[\/latex] and [latex]\\left(2,8\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve the system of nonlinear equations.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}{x}^{2}+{y}^{2}=10\\hfill \\\\ x - 3y=-10\\hfill \\end{array}[\/latex]<\/div>\r\n[reveal-answer q=\"369213\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"369213\"][latex]\\left(-1,3\\right)[\/latex][\/hidden-answer]\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-abfagheg-uflFybl5qyM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/uflFybl5qyM?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-abfagheg-uflFybl5qyM\" 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=12851024&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-abfagheg-uflFybl5qyM&amp;vembed=0&amp;video_id=uflFybl5qyM&amp;video_target=tpm-plugin-abfagheg-uflFybl5qyM\" 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\/Solve+a+NonLinear+System+of+Equations+(Linear+and+Quadratic)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolve a NonLinear System of Equations (Linear and Quadratic)\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-hhagabcg-VI4V_X_ZZLc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/VI4V_X_ZZLc?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hhagabcg-VI4V_X_ZZLc\" 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=12851025&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hhagabcg-VI4V_X_ZZLc&amp;vembed=0&amp;video_id=VI4V_X_ZZLc&amp;video_target=tpm-plugin-hhagabcg-VI4V_X_ZZLc\" 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+Nonlinear+Equations+(Substitution)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Solve a System of Nonlinear Equations (Substitution)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Solving a System of Nonlinear Equations Using 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\">Method Selection\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Elimination preferred when both equations are nonlinear<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Especially useful when equations have similar terms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">More efficient than substitution for two-by-two systems<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Works well with conic sections (circles, ellipses, etc.)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solution Types for Conic Intersections Circle-Ellipse Intersections can have:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">No solutions (no intersection points)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">One solution (curves are tangent)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Two solutions (curves intersect twice)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Three solutions (curves intersect three times)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Four solutions (curves intersect four times)<\/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 like terms between equations<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiply equations to match coefficients<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Add\/subtract to eliminate one variable<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve resulting single-variable equation<\/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\">Key Considerations\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Don't need to recognize curve types to solve<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Pay extra attention to algebra if curves unfamiliar<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Watch for extraneous solutions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check all solutions in both equations<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Use an online graphing calculator to find the solution to the system of equations.\r\n<div style=\"text-align: center;\">[latex]\\begin{gathered}4{x}^{2}+{y}^{2}=13\\\\ {x}^{2}+{y}^{2}=10\\end{gathered}[\/latex]<\/div>\r\n[reveal-answer q=\"880387\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"880387\"][latex]\\left\\{\\left(1,3\\right),\\left(1,-3\\right),\\left(-1,3\\right),\\left(-1,-3\\right)\\right\\}[\/latex][\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video, we present an example of how to solve a system of non-linear equations that represent the intersection of an ellipse and a hyperbola.\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-aecdbchh-Ic-42kmdJqc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Ic-42kmdJqc?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-aecdbchh-Ic-42kmdJqc\" 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=12851026&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-aecdbchh-Ic-42kmdJqc&amp;vembed=0&amp;video_id=Ic-42kmdJqc&amp;video_target=tpm-plugin-aecdbchh-Ic-42kmdJqc\" 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+3+-+Solve+a+System+of+Nonlinear+Equations+(Elimination)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 3: Solve a System of Nonlinear Equations (Elimination)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Graphing Inequalities<\/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\">Universal Steps for Inequality Graphing\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Convert inequality to equation by replacing inequality symbol<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graph boundary line\/curve<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use solid line for [latex]\\leq[\/latex] or [latex]\\geq[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use dashed line for [latex]\\lt[\/latex] or [latex]\\gt[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Test point to determine shading region<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Shade solution region<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Boundary Line Types\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Linear: [latex]ax + by \\leq c[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Parabolic: [latex]y \\leq ax^2 + bx + c[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Circle: [latex]x^2 + y^2 \\leq r^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Other nonlinear curves<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Region Testing Strategy\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Choose point NOT on boundary<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Pick simple point (like [latex](0,0)[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Test in original inequality<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If true, shade that region<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If false, shade opposite region<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Visual Indicators\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Solid line: includes boundary points<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Dashed line: excludes boundary points<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Shaded region: all solution points<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiple regions possible with systems<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Graphing a System of Nonlinear Inequalities<\/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 Components\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">At least one nonlinear inequality<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Two or more inequalities<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solution is intersection of regions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Feasible region where all inequalities are satisfied<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solution Process\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Find intersection points algebraically<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graph boundary curves<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Test regions for each inequality<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identify overlapping solution region<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Types of Boundaries\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Parabolas: [latex]y \\leq ax^2 + bx + c[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Circles: [latex]x^2 + y^2 \\leq r^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Other conics (ellipses, hyperbolas)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Mix of linear and nonlinear<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Region Analysis\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">May have multiple solution regions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Regions can be bounded\/unbounded<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Some regions may be disconnected<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check entire boundary for intersections<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Graph the given system of inequalities.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}y\\ge {x}^{2}-1 \\\\ x-y\\ge -1 \\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"689015\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"689015\"]\r\n\r\nShade the area bounded by the two curves, above the quadratic and below the line.\r\n\r\n[caption id=\"attachment_3175\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-3175 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/19182112\/CNX_Precalc_Figure_09_03_0122.jpg\" alt=\"A line intersecting a parabola at the points negative one, zero and two, three. The region under the line but above the parabola is shaded.\" width=\"487\" height=\"442\" \/> A line intersecting a parabola at the points (-1,0) and (2,3) with the area in between shaded[\/caption]\r\n\r\n[\/hidden-answer]\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-bfbcacgg-MifKjCFeTG0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/MifKjCFeTG0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bfbcacgg-MifKjCFeTG0\" 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=12851027&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bfbcacgg-MifKjCFeTG0&amp;vembed=0&amp;video_id=MifKjCFeTG0&amp;video_target=tpm-plugin-bfbcacgg-MifKjCFeTG0\" 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\/Graph+the+Solution+to+a+System+of+Inequalities.+(Quadratic%3ALinear)+Bounded_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraph the Solution to a System of Inequalities. (Quadratic\/Linear) Bounded\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-ccfcafbc-tzuK_c2oKI0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/tzuK_c2oKI0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ccfcafbc-tzuK_c2oKI0\" 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=12851028&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ccfcafbc-tzuK_c2oKI0&amp;vembed=0&amp;video_id=tzuK_c2oKI0&amp;video_target=tpm-plugin-ccfcafbc-tzuK_c2oKI0\" 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\/Graph+the+Solution+to+a+System+of+Inequalities.+(Quadratic%3ALinear)+No+Solution_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraph the Solution to a System of Inequalities. (Quadratic\/Linear) No Solution\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Solve a system of nonlinear equations.<\/li>\n<li>Graph a system of nonlinear inequalities.<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Solving a System of Nonlinear Equations Using Substitution<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p class=\"font-600 text-lg font-bold\"><strong>Common Features<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Solution Types\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">No solutions (no intersection)<\/li>\n<li class=\"whitespace-normal break-words\">One solution (tangent point)<\/li>\n<li class=\"whitespace-normal break-words\">Two solutions (line crosses through curve)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solution Method\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Always start with linear equation<\/li>\n<li class=\"whitespace-normal break-words\">Solve linear equation for one variable<\/li>\n<li class=\"whitespace-normal break-words\">Substitute into nonlinear equation<\/li>\n<li class=\"whitespace-normal break-words\">Solve resulting equation<\/li>\n<li class=\"whitespace-normal break-words\">Check all solutions in both equations<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p class=\"font-600 text-lg font-bold\"><strong>Line-Parabola Systems<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Form\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Line: [latex]ax + by = c[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Parabola: [latex]y = ax^2 + bx + c[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solution Process\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Results in quadratic equation<\/li>\n<li class=\"whitespace-normal break-words\">Two solutions: Line crosses parabola<\/li>\n<li class=\"whitespace-normal break-words\">One solution: Line tangent to parabola<\/li>\n<li class=\"whitespace-normal break-words\">No solutions: Line misses parabola<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p class=\"font-600 text-lg font-bold\"><strong>Line-Circle Systems<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Form\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Line: [latex]ax + by = c[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Circle: [latex](x - h)^2 + (y - k)^2 = r^2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solution Process\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Results in quadratic equation<\/li>\n<li class=\"whitespace-normal break-words\">Two solutions: Line crosses circle<\/li>\n<li class=\"whitespace-normal break-words\">One solution: Line tangent to circle<\/li>\n<li class=\"whitespace-normal break-words\">No solutions: Line outside circle<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the given system of equations by substitution.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{gathered}3x-y=-2 \\\\ 2{x}^{2}-y=0 \\end{gathered}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q503330\">Show Solution<\/button><\/p>\n<div id=\"q503330\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(-\\dfrac{1}{2},\\dfrac{1}{2}\\right)[\/latex] and [latex]\\left(2,8\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the system of nonlinear equations.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}{x}^{2}+{y}^{2}=10\\hfill \\\\ x - 3y=-10\\hfill \\end{array}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q369213\">Show Solution<\/button><\/p>\n<div id=\"q369213\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left(-1,3\\right)[\/latex]<\/div>\n<\/div>\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-abfagheg-uflFybl5qyM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/uflFybl5qyM?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-abfagheg-uflFybl5qyM\" 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=12851024&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-abfagheg-uflFybl5qyM&amp;vembed=0&amp;video_id=uflFybl5qyM&amp;video_target=tpm-plugin-abfagheg-uflFybl5qyM\" 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\/Solve+a+NonLinear+System+of+Equations+(Linear+and+Quadratic)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolve a NonLinear System of Equations (Linear and Quadratic)\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-hhagabcg-VI4V_X_ZZLc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/VI4V_X_ZZLc?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hhagabcg-VI4V_X_ZZLc\" 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=12851025&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hhagabcg-VI4V_X_ZZLc&amp;vembed=0&amp;video_id=VI4V_X_ZZLc&amp;video_target=tpm-plugin-hhagabcg-VI4V_X_ZZLc\" 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+Nonlinear+Equations+(Substitution)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Solve a System of Nonlinear Equations (Substitution)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Solving a System of Nonlinear Equations Using 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\">Method Selection\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Elimination preferred when both equations are nonlinear<\/li>\n<li class=\"whitespace-normal break-words\">Especially useful when equations have similar terms<\/li>\n<li class=\"whitespace-normal break-words\">More efficient than substitution for two-by-two systems<\/li>\n<li class=\"whitespace-normal break-words\">Works well with conic sections (circles, ellipses, etc.)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solution Types for Conic Intersections Circle-Ellipse Intersections can have:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">No solutions (no intersection points)<\/li>\n<li class=\"whitespace-normal break-words\">One solution (curves are tangent)<\/li>\n<li class=\"whitespace-normal break-words\">Two solutions (curves intersect twice)<\/li>\n<li class=\"whitespace-normal break-words\">Three solutions (curves intersect three times)<\/li>\n<li class=\"whitespace-normal break-words\">Four solutions (curves intersect four times)<\/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 like terms between equations<\/li>\n<li class=\"whitespace-normal break-words\">Multiply equations to match coefficients<\/li>\n<li class=\"whitespace-normal break-words\">Add\/subtract to eliminate one variable<\/li>\n<li class=\"whitespace-normal break-words\">Solve resulting single-variable equation<\/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\">Key Considerations\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Don&#8217;t need to recognize curve types to solve<\/li>\n<li class=\"whitespace-normal break-words\">Pay extra attention to algebra if curves unfamiliar<\/li>\n<li class=\"whitespace-normal break-words\">Watch for extraneous solutions<\/li>\n<li class=\"whitespace-normal break-words\">Check all solutions in both equations<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Use an online graphing calculator to find the solution to the system of equations.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{gathered}4{x}^{2}+{y}^{2}=13\\\\ {x}^{2}+{y}^{2}=10\\end{gathered}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q880387\">Show Solution<\/button><\/p>\n<div id=\"q880387\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left\\{\\left(1,3\\right),\\left(1,-3\\right),\\left(-1,3\\right),\\left(-1,-3\\right)\\right\\}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video, we present an example of how to solve a system of non-linear equations that represent the intersection of an ellipse and a hyperbola.<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-aecdbchh-Ic-42kmdJqc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Ic-42kmdJqc?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-aecdbchh-Ic-42kmdJqc\" 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=12851026&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-aecdbchh-Ic-42kmdJqc&amp;vembed=0&amp;video_id=Ic-42kmdJqc&amp;video_target=tpm-plugin-aecdbchh-Ic-42kmdJqc\" 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+3+-+Solve+a+System+of+Nonlinear+Equations+(Elimination)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 3: Solve a System of Nonlinear Equations (Elimination)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Graphing Inequalities<\/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\">Universal Steps for Inequality Graphing\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Convert inequality to equation by replacing inequality symbol<\/li>\n<li class=\"whitespace-normal break-words\">Graph boundary line\/curve<\/li>\n<li class=\"whitespace-normal break-words\">Use solid line for [latex]\\leq[\/latex] or [latex]\\geq[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Use dashed line for [latex]\\lt[\/latex] or [latex]\\gt[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Test point to determine shading region<\/li>\n<li class=\"whitespace-normal break-words\">Shade solution region<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Boundary Line Types\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Linear: [latex]ax + by \\leq c[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Parabolic: [latex]y \\leq ax^2 + bx + c[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Circle: [latex]x^2 + y^2 \\leq r^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Other nonlinear curves<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Region Testing Strategy\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Choose point NOT on boundary<\/li>\n<li class=\"whitespace-normal break-words\">Pick simple point (like [latex](0,0)[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Test in original inequality<\/li>\n<li class=\"whitespace-normal break-words\">If true, shade that region<\/li>\n<li class=\"whitespace-normal break-words\">If false, shade opposite region<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Visual Indicators\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Solid line: includes boundary points<\/li>\n<li class=\"whitespace-normal break-words\">Dashed line: excludes boundary points<\/li>\n<li class=\"whitespace-normal break-words\">Shaded region: all solution points<\/li>\n<li class=\"whitespace-normal break-words\">Multiple regions possible with systems<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2>Graphing a System of Nonlinear Inequalities<\/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 Components\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">At least one nonlinear inequality<\/li>\n<li class=\"whitespace-normal break-words\">Two or more inequalities<\/li>\n<li class=\"whitespace-normal break-words\">Solution is intersection of regions<\/li>\n<li class=\"whitespace-normal break-words\">Feasible region where all inequalities are satisfied<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solution Process\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Find intersection points algebraically<\/li>\n<li class=\"whitespace-normal break-words\">Graph boundary curves<\/li>\n<li class=\"whitespace-normal break-words\">Test regions for each inequality<\/li>\n<li class=\"whitespace-normal break-words\">Identify overlapping solution region<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Types of Boundaries\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Parabolas: [latex]y \\leq ax^2 + bx + c[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Circles: [latex]x^2 + y^2 \\leq r^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Other conics (ellipses, hyperbolas)<\/li>\n<li class=\"whitespace-normal break-words\">Mix of linear and nonlinear<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Region Analysis\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">May have multiple solution regions<\/li>\n<li class=\"whitespace-normal break-words\">Regions can be bounded\/unbounded<\/li>\n<li class=\"whitespace-normal break-words\">Some regions may be disconnected<\/li>\n<li class=\"whitespace-normal break-words\">Check entire boundary for intersections<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Graph the given system of inequalities.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}y\\ge {x}^{2}-1 \\\\ x-y\\ge -1 \\end{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q689015\">Show Solution<\/button><\/p>\n<div id=\"q689015\" class=\"hidden-answer\" style=\"display: none\">\n<p>Shade the area bounded by the two curves, above the quadratic and below the line.<\/p>\n<figure id=\"attachment_3175\" aria-describedby=\"caption-attachment-3175\" style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3175 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/19182112\/CNX_Precalc_Figure_09_03_0122.jpg\" alt=\"A line intersecting a parabola at the points negative one, zero and two, three. The region under the line but above the parabola is shaded.\" width=\"487\" height=\"442\" \/><figcaption id=\"caption-attachment-3175\" class=\"wp-caption-text\">A line intersecting a parabola at the points (-1,0) and (2,3) with the area in between shaded<\/figcaption><\/figure>\n<\/div>\n<\/div>\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-bfbcacgg-MifKjCFeTG0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/MifKjCFeTG0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bfbcacgg-MifKjCFeTG0\" 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=12851027&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bfbcacgg-MifKjCFeTG0&amp;vembed=0&amp;video_id=MifKjCFeTG0&amp;video_target=tpm-plugin-bfbcacgg-MifKjCFeTG0\" 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\/Graph+the+Solution+to+a+System+of+Inequalities.+(Quadratic%3ALinear)+Bounded_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraph the Solution to a System of Inequalities. (Quadratic\/Linear) Bounded\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-ccfcafbc-tzuK_c2oKI0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/tzuK_c2oKI0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ccfcafbc-tzuK_c2oKI0\" 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=12851028&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ccfcafbc-tzuK_c2oKI0&amp;vembed=0&amp;video_id=tzuK_c2oKI0&amp;video_target=tpm-plugin-ccfcafbc-tzuK_c2oKI0\" 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\/Graph+the+Solution+to+a+System+of+Inequalities.+(Quadratic%3ALinear)+No+Solution_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraph the Solution to a System of Inequalities. (Quadratic\/Linear) No Solution\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":26,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Solve a NonLinear System of Equations (Linear and Quadratic)\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/uflFybl5qyM\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 1: Solve a System of Nonlinear Equations (Substitution)\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/VI4V_X_ZZLc\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 3: Solve a System of Nonlinear Equations (Elimination)\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/Ic-42kmdJqc\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Graph the Solution to a System of Inequalities. 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