{"id":2656,"date":"2025-08-13T18:21:43","date_gmt":"2025-08-13T18:21:43","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2656"},"modified":"2026-02-24T21:26:19","modified_gmt":"2026-02-24T21:26:19","slug":"systems-of-equations-and-inequalities-background-youll-need-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/systems-of-equations-and-inequalities-background-youll-need-5\/","title":{"raw":"Systems of Equations and Inequalities: Background You'll Need 5","rendered":"Systems of Equations and Inequalities: Background You&#8217;ll Need 5"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Graph horizontal and vertical lines<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>horizontal and vertical lines<\/h3>\r\n<ul>\r\n \t<li>Horizontal lines:\r\n<ul>\r\n \t<li>Equation: [latex]y=c[\/latex].<\/li>\r\n \t<li>Slope [latex]0[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Vertical lines:\r\n<ul>\r\n \t<li>Equation: [latex]x=c[\/latex].<\/li>\r\n \t<li>Slope undefined.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Graph each line.\r\n<ol>\r\n \t<li>[latex]y=-3[\/latex]<\/li>\r\n \t<li>[latex]x=4[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"838372\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"838372\"]\r\n<ol>\r\n \t<li>[latex]y=-3[\/latex]\r\nAll points on this line have a [latex]y[\/latex]-coordinate of [latex]-3[\/latex]. We can generate points to plot:\r\n[latex](0,-3)[\/latex], [latex](3,-3)[\/latex], [latex](-2,-3)[\/latex]\r\n<img class=\"alignnone wp-image-4078\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/17162216\/Screenshot-2025-09-17-at-9.22.02%E2%80%AFAM.png\" alt=\"Horizontal line crosses the y-axis at negative 3.\" width=\"245\" height=\"237\" \/><\/li>\r\n \t<li>[latex]x=4[\/latex]\r\nAll points on this line have an [latex]x[\/latex]-coordinate of [latex]4[\/latex]. We can generate points to plot:\r\n[latex](4,0)[\/latex], [latex](4,-3)[\/latex], [latex](4,2)[\/latex]\r\n<img class=\"alignnone wp-image-4079\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/17162542\/Screenshot-2025-09-17-at-9.25.31%E2%80%AFAM.png\" alt=\"Vertical line crosses the x-axis at 4.\" width=\"233\" height=\"252\" \/><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]312254[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Graph horizontal and vertical lines<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>horizontal and vertical lines<\/h3>\n<ul>\n<li>Horizontal lines:\n<ul>\n<li>Equation: [latex]y=c[\/latex].<\/li>\n<li>Slope [latex]0[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li>Vertical lines:\n<ul>\n<li>Equation: [latex]x=c[\/latex].<\/li>\n<li>Slope undefined.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Graph each line.<\/p>\n<ol>\n<li>[latex]y=-3[\/latex]<\/li>\n<li>[latex]x=4[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q838372\">Show Solution<\/button><\/p>\n<div id=\"q838372\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]y=-3[\/latex]<br \/>\nAll points on this line have a [latex]y[\/latex]-coordinate of [latex]-3[\/latex]. We can generate points to plot:<br \/>\n[latex](0,-3)[\/latex], [latex](3,-3)[\/latex], [latex](-2,-3)[\/latex]<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-4078\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/17162216\/Screenshot-2025-09-17-at-9.22.02%E2%80%AFAM.png\" alt=\"Horizontal line crosses the y-axis at negative 3.\" width=\"245\" height=\"237\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/17162216\/Screenshot-2025-09-17-at-9.22.02%E2%80%AFAM.png 726w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/17162216\/Screenshot-2025-09-17-at-9.22.02%E2%80%AFAM-300x291.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/17162216\/Screenshot-2025-09-17-at-9.22.02%E2%80%AFAM-65x63.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/17162216\/Screenshot-2025-09-17-at-9.22.02%E2%80%AFAM-225x218.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/17162216\/Screenshot-2025-09-17-at-9.22.02%E2%80%AFAM-350x339.png 350w\" sizes=\"(max-width: 245px) 100vw, 245px\" \/><\/li>\n<li>[latex]x=4[\/latex]<br \/>\nAll points on this line have an [latex]x[\/latex]-coordinate of [latex]4[\/latex]. We can generate points to plot:<br \/>\n[latex](4,0)[\/latex], [latex](4,-3)[\/latex], [latex](4,2)[\/latex]<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-4079\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/17162542\/Screenshot-2025-09-17-at-9.25.31%E2%80%AFAM.png\" alt=\"Vertical line crosses the x-axis at 4.\" width=\"233\" height=\"252\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/17162542\/Screenshot-2025-09-17-at-9.25.31%E2%80%AFAM.png 736w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/17162542\/Screenshot-2025-09-17-at-9.25.31%E2%80%AFAM-277x300.png 277w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/17162542\/Screenshot-2025-09-17-at-9.25.31%E2%80%AFAM-65x70.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/17162542\/Screenshot-2025-09-17-at-9.25.31%E2%80%AFAM-225x243.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/17162542\/Screenshot-2025-09-17-at-9.25.31%E2%80%AFAM-350x379.png 350w\" sizes=\"(max-width: 233px) 100vw, 233px\" \/><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm312254\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312254&theme=lumen&iframe_resize_id=ohm312254&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":67,"menu_order":6,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":131,"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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2656"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2656\/revisions"}],"predecessor-version":[{"id":5729,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2656\/revisions\/5729"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/131"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2656\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2656"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2656"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2656"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2656"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}