{"id":2511,"date":"2025-08-13T17:21:30","date_gmt":"2025-08-13T17:21:30","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2511"},"modified":"2026-01-09T20:23:13","modified_gmt":"2026-01-09T20:23:13","slug":"linear-functions-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/linear-functions-background-youll-need-2\/","title":{"raw":"Linear Functions: Background You'll Need 2","rendered":"Linear Functions: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Recognize parallel and perpendicular lines<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Parallel and Perpendicular Lines<\/h2>\r\nParallel lines are a key idea in geometry. They are lines that stay the same distance apart over their entire length and never cross each other, no matter how long they extend. This consistent distance between them is called <strong>equidistance<\/strong>. In simpler terms, no matter how far you extend parallel lines, they will always run alongside each other and never meet. This makes them an important concept to understand, especially when studying shapes, angles, and various design and engineering principles.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"386\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/17232102\/CNX_CAT_Figure_02_02_004.jpg\" alt=\"Coordinate plane with the x-axis ranging from negative 8 to 8 in intervals of 2 and the y-axis ranging from negative 7 to 7. Three functions are graphed on the same plot: y = 2 times x minus 3; y = 2 times x plus 1 and y = 2 times x plus 5.\" width=\"386\" height=\"470\" \/> Parallel lines have slopes that are the same.[\/caption]\r\n\r\nAll of the lines shown in the graph are parallel because they have the same slope and different <em>y-<\/em>intercepts.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>parallel lines<\/h3>\r\n<strong>Parallel lines<\/strong> are defined as lines in a plane that do not intersect because they have the same slope, maintaining a consistent direction and steepness.\r\n\r\n<\/section>Lines that are <strong>perpendicular<\/strong> intersect to form a [latex]{90}^{\\circ }[\/latex] angle. The slope of one line is the <strong>negative<\/strong> <strong>reciprocal<\/strong> of the other. We can show that two lines are perpendicular if the product of the two slopes is [latex]-1:{m}_{1}\\cdot {m}_{2}=-1[\/latex].\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">For example, the figure below shows the graph of two perpendicular lines. One line has a slope of [latex]3[\/latex]; the other line has a slope of [latex]-\\dfrac{1}{3}[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{ }{m}_{1}\\cdot {m}_{2}=-1\\hfill \\\\ \\text{ }3\\cdot \\left(-\\dfrac{1}{3}\\right)=-1\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/17232106\/CNX_CAT_Figure_02_02_005.jpg\" alt=\"Coordinate plane with the x-axis ranging from negative 3 to 6 and the y-axis ranging from negative 2 to 5. Two functions are graphed on the same plot: y = 3 times x minus 1 and y = negative x\/3 minus 2. Their intersection is marked by a box to show that it is a right angle.\" width=\"487\" height=\"329\" \/> Perpendicular lines have slopes that are negative reciprocals of each other.[\/caption]\r\n\r\n<\/section><section class=\"textbox keyTakeaway\">\r\n<h3>perpendicular lines<\/h3>\r\nLines that are <strong>perpendicular<\/strong> intersect to form a [latex]{90}^{\\circ }[\/latex] angle.\r\n\r\n&nbsp;\r\n\r\nThis relationship occurs when the slopes of two lines are negative reciprocals of each other, meaning if one line has a slope of [latex]m[\/latex], the perpendicular line will have a slope of [latex]-\\dfrac{1}{m}[\/latex].\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]318702[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Recognize parallel and perpendicular lines<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Parallel and Perpendicular Lines<\/h2>\n<p>Parallel lines are a key idea in geometry. They are lines that stay the same distance apart over their entire length and never cross each other, no matter how long they extend. This consistent distance between them is called <strong>equidistance<\/strong>. In simpler terms, no matter how far you extend parallel lines, they will always run alongside each other and never meet. This makes them an important concept to understand, especially when studying shapes, angles, and various design and engineering principles.<\/p>\n<p>&nbsp;<\/p>\n<figure style=\"width: 386px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/17232102\/CNX_CAT_Figure_02_02_004.jpg\" alt=\"Coordinate plane with the x-axis ranging from negative 8 to 8 in intervals of 2 and the y-axis ranging from negative 7 to 7. Three functions are graphed on the same plot: y = 2 times x minus 3; y = 2 times x plus 1 and y = 2 times x plus 5.\" width=\"386\" height=\"470\" \/><figcaption class=\"wp-caption-text\">Parallel lines have slopes that are the same.<\/figcaption><\/figure>\n<p>All of the lines shown in the graph are parallel because they have the same slope and different <em>y-<\/em>intercepts.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>parallel lines<\/h3>\n<p><strong>Parallel lines<\/strong> are defined as lines in a plane that do not intersect because they have the same slope, maintaining a consistent direction and steepness.<\/p>\n<\/section>\n<p>Lines that are <strong>perpendicular<\/strong> intersect to form a [latex]{90}^{\\circ }[\/latex] angle. The slope of one line is the <strong>negative<\/strong> <strong>reciprocal<\/strong> of the other. We can show that two lines are perpendicular if the product of the two slopes is [latex]-1:{m}_{1}\\cdot {m}_{2}=-1[\/latex].<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">For example, the figure below shows the graph of two perpendicular lines. One line has a slope of [latex]3[\/latex]; the other line has a slope of [latex]-\\dfrac{1}{3}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{ }{m}_{1}\\cdot {m}_{2}=-1\\hfill \\\\ \\text{ }3\\cdot \\left(-\\dfrac{1}{3}\\right)=-1\\hfill \\end{array}[\/latex]<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/17232106\/CNX_CAT_Figure_02_02_005.jpg\" alt=\"Coordinate plane with the x-axis ranging from negative 3 to 6 and the y-axis ranging from negative 2 to 5. Two functions are graphed on the same plot: y = 3 times x minus 1 and y = negative x\/3 minus 2. Their intersection is marked by a box to show that it is a right angle.\" width=\"487\" height=\"329\" \/><figcaption class=\"wp-caption-text\">Perpendicular lines have slopes that are negative reciprocals of each other.<\/figcaption><\/figure>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>perpendicular lines<\/h3>\n<p>Lines that are <strong>perpendicular<\/strong> intersect to form a [latex]{90}^{\\circ }[\/latex] angle.<\/p>\n<p>&nbsp;<\/p>\n<p>This relationship occurs when the slopes of two lines are negative reciprocals of each other, meaning if one line has a slope of [latex]m[\/latex], the perpendicular line will have a slope of [latex]-\\dfrac{1}{m}[\/latex].<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm318702\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318702&theme=lumen&iframe_resize_id=ohm318702&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":67,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":61,"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\/2511"}],"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\/2511\/revisions"}],"predecessor-version":[{"id":5259,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2511\/revisions\/5259"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/61"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2511\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2511"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2511"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2511"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2511"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}