{"id":2508,"date":"2025-08-13T17:21:06","date_gmt":"2025-08-13T17:21:06","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2508"},"modified":"2026-01-09T19:46:05","modified_gmt":"2026-01-09T19:46:05","slug":"linear-functions-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/linear-functions-background-youll-need-1\/","title":{"raw":"Linear Functions: Background You'll Need 1","rendered":"Linear Functions: 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\">Find the slope of a line and explain what it means<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Finding the Slope of a Line<\/h2>\r\nThe slope of a line tells us how steep the line is and which direction it goes. It's like measuring how much you go up (or down) for every step you take to the right.\r\n\r\n<section class=\"textbox recall\">\r\n<p style=\"text-align: center;\">[latex]m =\\dfrac{\\text{rise}}{\\text{run}} = \\dfrac{y_2 - y_1}{x_2 - x_1}[\/latex]<\/p>\r\nWhere:\r\n<ul>\r\n \t<li>[latex](x_1, y_1)[\/latex] and [latex](x_2, y_2)[\/latex] are two points on the line.<\/li>\r\n \t<li>[latex]m[\/latex] is the slope of the line.<\/li>\r\n \t<li>\"rise\" is the change in output.<\/li>\r\n \t<li>\"run\" is the change in input.<\/li>\r\n<\/ul>\r\nWhen interpreting slope, it will be important to consider the units of measurement. Make sure to always attach these units to both the numerator and denominator when they are provided to you.\r\n\r\n[caption id=\"attachment_1780\" align=\"aligncenter\" width=\"294\"]<img class=\"wp-image-1780 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/06205708\/Screenshot-2024-06-06-at-1.57.04%E2%80%AFPM.png\" alt=\"\" width=\"294\" height=\"141\" \/> Guide on slopes of lines[\/caption]\r\n\r\n<\/section><section class=\"textbox example\">\r\n\r\n[caption id=\"attachment_1781\" align=\"alignright\" width=\"222\"]<img class=\"wp-image-1781 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/06205744\/Screenshot-2024-06-06-at-1.57.26%E2%80%AFPM.png\" alt=\"\" width=\"222\" height=\"169\" \/> Graph of a line[\/caption]\r\n\r\nFind the slope of the line shown.\r\n\r\n[reveal-answer q=\"639188\"]Using two points[\/reveal-answer]\r\n[hidden-answer a=\"639188\"]\r\n\r\nLocate two points on the graph whose coordinates are integers. Label the coordinates of these points.\r\n<ul>\r\n \t<li>[latex](0,1) = (x_1, y_1)[\/latex]<\/li>\r\n \t<li>[latex](5, -2) = (x_2, y_2)[\/latex]<\/li>\r\n<\/ul>\r\n[latex]\\begin{align*} m &amp;= \\dfrac{y_2 - y_1}{x_2 - x_1} \\\\ &amp;= \\dfrac{-2 - 1}{5 - 0} \\\\ &amp;= \\dfrac{-3}{5} \\end{align*}[\/latex]\r\n\r\nSo, the slope of the line is [latex]-\\dfrac{3}{5}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"752920\"]Using graph (rise and run)[\/reveal-answer]\r\n[hidden-answer a=\"752920\"]<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">It may help to visualize this change as [latex]m =\\dfrac{\\text{rise}}{\\text{run}}[\/latex].\u00a0<\/span>\r\n\r\nCount the rise between the points. Since the line goes down, the slope is negative. <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Then count the run, or horizontal change.<\/span>\r\n\r\n[caption id=\"attachment_1782\" align=\"aligncenter\" width=\"222\"]<img class=\"wp-image-1782 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/06210334\/Screenshot-2024-06-06-at-1.57.26%E2%80%AFPM-1.png\" alt=\"\" width=\"222\" height=\"169\" \/> Graph of a line with slope labels[\/caption]\r\n\r\nSo, the slope of the line is <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]m =\\dfrac{\\text{rise}}{\\text{run}} = \\dfrac{-3}{5}[\/latex].\u00a0<\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">Find the slope of [latex]2x-4y = 5[\/latex].[reveal-answer q=\"525438\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"525438\"]To find the slope of the line given by the equation [latex]2x-4y = 5[\/latex], we need to rewrite the equation in slope-intercept form [latex]y = mx+b[\/latex], where [latex]m[\/latex] is the slope.<center>[latex]\\begin{align*} 2x - 4y &amp;= 5 \\\\ -4y &amp;= -2x + 5 \\\\ y &amp;= \\frac{-2x + 5}{-4} \\\\ y &amp;= \\frac{-2x}{-4} + \\frac{5}{-4} \\\\ y &amp;= \\frac{1}{2}x - \\frac{5}{4} \\end{align*}[\/latex]<\/center>\r\nSo, the slope of the line is [latex]\\dfrac{1}{2}[\/latex].[\/hidden-answer]<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]318700[\/ohm_question]<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]318701[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Find the slope of a line and explain what it means<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Finding the Slope of a Line<\/h2>\n<p>The slope of a line tells us how steep the line is and which direction it goes. It&#8217;s like measuring how much you go up (or down) for every step you take to the right.<\/p>\n<section class=\"textbox recall\">\n<p style=\"text-align: center;\">[latex]m =\\dfrac{\\text{rise}}{\\text{run}} = \\dfrac{y_2 - y_1}{x_2 - x_1}[\/latex]<\/p>\n<p>Where:<\/p>\n<ul>\n<li>[latex](x_1, y_1)[\/latex] and [latex](x_2, y_2)[\/latex] are two points on the line.<\/li>\n<li>[latex]m[\/latex] is the slope of the line.<\/li>\n<li>&#8220;rise&#8221; is the change in output.<\/li>\n<li>&#8220;run&#8221; is the change in input.<\/li>\n<\/ul>\n<p>When interpreting slope, it will be important to consider the units of measurement. Make sure to always attach these units to both the numerator and denominator when they are provided to you.<\/p>\n<figure id=\"attachment_1780\" aria-describedby=\"caption-attachment-1780\" style=\"width: 294px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1780 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/06205708\/Screenshot-2024-06-06-at-1.57.04%E2%80%AFPM.png\" alt=\"\" width=\"294\" height=\"141\" \/><figcaption id=\"caption-attachment-1780\" class=\"wp-caption-text\">Guide on slopes of lines<\/figcaption><\/figure>\n<\/section>\n<section class=\"textbox example\">\n<figure id=\"attachment_1781\" aria-describedby=\"caption-attachment-1781\" style=\"width: 222px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1781 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/06205744\/Screenshot-2024-06-06-at-1.57.26%E2%80%AFPM.png\" alt=\"\" width=\"222\" height=\"169\" \/><figcaption id=\"caption-attachment-1781\" class=\"wp-caption-text\">Graph of a line<\/figcaption><\/figure>\n<p>Find the slope of the line shown.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q639188\">Using two points<\/button><\/p>\n<div id=\"q639188\" class=\"hidden-answer\" style=\"display: none\">\n<p>Locate two points on the graph whose coordinates are integers. Label the coordinates of these points.<\/p>\n<ul>\n<li>[latex](0,1) = (x_1, y_1)[\/latex]<\/li>\n<li>[latex](5, -2) = (x_2, y_2)[\/latex]<\/li>\n<\/ul>\n<p>[latex]\\begin{align*} m &= \\dfrac{y_2 - y_1}{x_2 - x_1} \\\\ &= \\dfrac{-2 - 1}{5 - 0} \\\\ &= \\dfrac{-3}{5} \\end{align*}[\/latex]<\/p>\n<p>So, the slope of the line is [latex]-\\dfrac{3}{5}[\/latex].<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q752920\">Using graph (rise and run)<\/button><\/p>\n<div id=\"q752920\" class=\"hidden-answer\" style=\"display: none\"><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">It may help to visualize this change as [latex]m =\\dfrac{\\text{rise}}{\\text{run}}[\/latex].\u00a0<\/span><\/p>\n<p>Count the rise between the points. Since the line goes down, the slope is negative. <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Then count the run, or horizontal change.<\/span><\/p>\n<figure id=\"attachment_1782\" aria-describedby=\"caption-attachment-1782\" style=\"width: 222px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1782 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/06210334\/Screenshot-2024-06-06-at-1.57.26%E2%80%AFPM-1.png\" alt=\"\" width=\"222\" height=\"169\" \/><figcaption id=\"caption-attachment-1782\" class=\"wp-caption-text\">Graph of a line with slope labels<\/figcaption><\/figure>\n<p>So, the slope of the line is <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]m =\\dfrac{\\text{rise}}{\\text{run}} = \\dfrac{-3}{5}[\/latex].\u00a0<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Find the slope of [latex]2x-4y = 5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q525438\">Show Answer<\/button><\/p>\n<div id=\"q525438\" class=\"hidden-answer\" style=\"display: none\">To find the slope of the line given by the equation [latex]2x-4y = 5[\/latex], we need to rewrite the equation in slope-intercept form [latex]y = mx+b[\/latex], where [latex]m[\/latex] is the slope.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align*} 2x - 4y &= 5 \\\\ -4y &= -2x + 5 \\\\ y &= \\frac{-2x + 5}{-4} \\\\ y &= \\frac{-2x}{-4} + \\frac{5}{-4} \\\\ y &= \\frac{1}{2}x - \\frac{5}{4} \\end{align*}[\/latex]<\/div>\n<p>So, the slope of the line is [latex]\\dfrac{1}{2}[\/latex].<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm318700\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318700&theme=lumen&iframe_resize_id=ohm318700&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm318701\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318701&theme=lumen&iframe_resize_id=ohm318701&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":67,"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":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\/2508"}],"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":6,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2508\/revisions"}],"predecessor-version":[{"id":5258,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2508\/revisions\/5258"}],"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\/2508\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2508"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2508"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2508"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2508"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}