{"id":2307,"date":"2024-07-22T20:44:54","date_gmt":"2024-07-22T20:44:54","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2307"},"modified":"2025-01-06T18:38:22","modified_gmt":"2025-01-06T18:38:22","slug":"module-13-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/module-13-background-youll-need-1\/","title":{"raw":"Systems of Equations and Inequalities: Background You'll Need 1","rendered":"Systems of Equations and Inequalities: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Check if a pair of numbers works as a solution for a set of equations<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"whitespace-pre-wrap break-words\">Determining if an Ordered Pair is a Solution to a System of Equations<\/h2>\r\nA system of equations consists of two or more equations with the same variables. An ordered pair [latex](x, y)[\/latex] is a solution to a system of equations if it satisfies all equations in the system simultaneously.\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p class=\"whitespace-pre-wrap break-words\"><strong>How to: Determine if an Ordered Pair is a Solution<\/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\">Substitute the [latex]x[\/latex] and [latex]y[\/latex] values of the ordered pair into both equations.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify each equation.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Verify if both equations are true.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Consider the system of equations:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{gathered}&amp; 2x + y = 10 \\\\ &amp; x - y = 2 \\end{gathered}[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Let's check if [latex](4, 2)[\/latex] is a solution.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">For equation 1: [latex]2x + y = 10[\/latex] Substitute [latex]x = 4[\/latex] and [latex]y = 2[\/latex]:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]2(4) + 2 = 10[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]8 + 2 = 10[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]10 = 10[\/latex] (True)<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">For equation 2: [latex]x - y = 2[\/latex] Substitute [latex]x = 4[\/latex] and [latex]y = 2[\/latex]:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]4 - 2 = 2[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]2 = 2[\/latex] (True)<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Since both equations are true when we substitute [latex](4, 2)[\/latex], this ordered pair is a solution to the system.<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Determine if the ordered pair [latex](3, -1)[\/latex] is a solution to the following system of equations:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{gathered}&amp; 3x - 2y = 11 \\\\ &amp; x + 4y = -1 \\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"494186\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"494186\"]\r\n<p class=\"whitespace-pre-wrap break-words\">For equation 1: [latex]3x - 2y = 11[\/latex] Substitute [latex]x = 3[\/latex] and [latex]y = -1[\/latex]:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]3(3) - 2(-1) = 11[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]9 + 2 = 11[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]11 = 11[\/latex] (True)<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">For equation 2: [latex]x + 4y = -1[\/latex] Substitute [latex]x = 3[\/latex] and [latex]y = -1[\/latex]:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]3 + 4(-1) = -1[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]3 - 4 = -1[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]-1 = -1[\/latex] (True)<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Since both equations are true when we substitute [latex](3, -1)[\/latex], this ordered pair is a solution to the system.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Determine if the ordered pair [latex](-2, 5)[\/latex] is a solution to the system:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{gathered}&amp; 2x + 3y = 11 \\\\ &amp; x - y = -7 \\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"783282\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"783282\"]\r\n<p class=\"whitespace-pre-wrap break-words\">For equation 1: [latex]2x + 3y = 11[\/latex] Substitute [latex]x = -2[\/latex] and [latex]y = 5[\/latex]:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]2(-2) + 3(5) = 11[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]-4 + 15 = 11[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]11 = 11[\/latex] (True)<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">For equation 2: [latex]x - y = -7[\/latex] Substitute [latex]x = -2[\/latex] and [latex]y = 5[\/latex]:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]-2 - 5 = -7[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]-7 = -7[\/latex] (True)<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Since both equations are true when we substitute [latex](-2, 5)[\/latex], this ordered pair is a solution to the system.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Determine if the ordered pair [latex](1, 2)[\/latex] is a solution to the system:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{gathered}&amp; 4x - y = 2 \\\\ &amp; x + 2y = 6 \\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"276709\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"276709\"]\r\n<p class=\"whitespace-pre-wrap break-words\">For equation 1: [latex]4x - y = 2[\/latex] Substitute [latex]x = 1[\/latex] and [latex]y = 2[\/latex]:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]4(1) - 2 = 2[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]4 - 2 = 2[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]2 = 2[\/latex] (True)<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">For equation 2: [latex]x + 2y = 6[\/latex] Substitute [latex]x = 1[\/latex] and [latex]y = 2[\/latex]:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]1 + 2(2) = 6[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]1 + 4 = 6[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]5 = 6[\/latex] (False)<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Although the ordered pair [latex](1, 2)[\/latex] satisfies the first equation, it does not satisfy the second equation. Therefore, [latex](1, 2)[\/latex] is not a solution to this system of equations.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]294608[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Check if a pair of numbers works as a solution for a set of equations<\/span><\/li>\n<\/ul>\n<\/section>\n<h2 class=\"whitespace-pre-wrap break-words\">Determining if an Ordered Pair is a Solution to a System of Equations<\/h2>\n<p>A system of equations consists of two or more equations with the same variables. An ordered pair [latex](x, y)[\/latex] is a solution to a system of equations if it satisfies all equations in the system simultaneously.<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p class=\"whitespace-pre-wrap break-words\"><strong>How to: Determine if an Ordered Pair is a Solution<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Substitute the [latex]x[\/latex] and [latex]y[\/latex] values of the ordered pair into both equations.<\/li>\n<li class=\"whitespace-normal break-words\">Simplify each equation.<\/li>\n<li class=\"whitespace-normal break-words\">Verify if both equations are true.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Consider the system of equations:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{gathered}& 2x + y = 10 \\\\ & x - y = 2 \\end{gathered}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Let&#8217;s check if [latex](4, 2)[\/latex] is a solution.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">For equation 1: [latex]2x + y = 10[\/latex] Substitute [latex]x = 4[\/latex] and [latex]y = 2[\/latex]:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]2(4) + 2 = 10[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]8 + 2 = 10[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]10 = 10[\/latex] (True)<\/p>\n<p class=\"whitespace-pre-wrap break-words\">For equation 2: [latex]x - y = 2[\/latex] Substitute [latex]x = 4[\/latex] and [latex]y = 2[\/latex]:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]4 - 2 = 2[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]2 = 2[\/latex] (True)<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Since both equations are true when we substitute [latex](4, 2)[\/latex], this ordered pair is a solution to the system.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Determine if the ordered pair [latex](3, -1)[\/latex] is a solution to the following system of equations:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{gathered}& 3x - 2y = 11 \\\\ & x + 4y = -1 \\end{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q494186\">Show Answer<\/button><\/p>\n<div id=\"q494186\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"whitespace-pre-wrap break-words\">For equation 1: [latex]3x - 2y = 11[\/latex] Substitute [latex]x = 3[\/latex] and [latex]y = -1[\/latex]:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]3(3) - 2(-1) = 11[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]9 + 2 = 11[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]11 = 11[\/latex] (True)<\/p>\n<p class=\"whitespace-pre-wrap break-words\">For equation 2: [latex]x + 4y = -1[\/latex] Substitute [latex]x = 3[\/latex] and [latex]y = -1[\/latex]:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]3 + 4(-1) = -1[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]3 - 4 = -1[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]-1 = -1[\/latex] (True)<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Since both equations are true when we substitute [latex](3, -1)[\/latex], this ordered pair is a solution to the system.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Determine if the ordered pair [latex](-2, 5)[\/latex] is a solution to the system:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{gathered}& 2x + 3y = 11 \\\\ & x - y = -7 \\end{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q783282\">Show Answer<\/button><\/p>\n<div id=\"q783282\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"whitespace-pre-wrap break-words\">For equation 1: [latex]2x + 3y = 11[\/latex] Substitute [latex]x = -2[\/latex] and [latex]y = 5[\/latex]:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]2(-2) + 3(5) = 11[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]-4 + 15 = 11[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]11 = 11[\/latex] (True)<\/p>\n<p class=\"whitespace-pre-wrap break-words\">For equation 2: [latex]x - y = -7[\/latex] Substitute [latex]x = -2[\/latex] and [latex]y = 5[\/latex]:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]-2 - 5 = -7[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]-7 = -7[\/latex] (True)<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Since both equations are true when we substitute [latex](-2, 5)[\/latex], this ordered pair is a solution to the system.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Determine if the ordered pair [latex](1, 2)[\/latex] is a solution to the system:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{gathered}& 4x - y = 2 \\\\ & x + 2y = 6 \\end{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q276709\">Show Answer<\/button><\/p>\n<div id=\"q276709\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"whitespace-pre-wrap break-words\">For equation 1: [latex]4x - y = 2[\/latex] Substitute [latex]x = 1[\/latex] and [latex]y = 2[\/latex]:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]4(1) - 2 = 2[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]4 - 2 = 2[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]2 = 2[\/latex] (True)<\/p>\n<p class=\"whitespace-pre-wrap break-words\">For equation 2: [latex]x + 2y = 6[\/latex] Substitute [latex]x = 1[\/latex] and [latex]y = 2[\/latex]:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]1 + 2(2) = 6[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]1 + 4 = 6[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]5 = 6[\/latex] (False)<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Although the ordered pair [latex](1, 2)[\/latex] satisfies the first equation, it does not satisfy the second equation. Therefore, [latex](1, 2)[\/latex] is not a solution to this system of equations.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm294608\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=294608&theme=lumen&iframe_resize_id=ohm294608&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":300,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2307"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":13,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2307\/revisions"}],"predecessor-version":[{"id":7048,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2307\/revisions\/7048"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/300"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2307\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=2307"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2307"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=2307"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=2307"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}