{"id":254,"date":"2023-02-20T17:14:05","date_gmt":"2023-02-20T17:14:05","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/equation-of-the-line-of-best-fit-background-youll-need-1\/"},"modified":"2025-05-11T23:07:49","modified_gmt":"2025-05-11T23:07:49","slug":"module-5-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/module-5-background-youll-need-1\/","title":{"raw":"Module 5: Background You'll Need 3","rendered":"Module 5: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Find and explain the slope and y-intercept of a linear model.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Linear Equation<\/h2>\r\n<section class=\"textbox recall\">The general form of a linear equation: [latex]y=mx+b[\/latex], where\u00a0[latex]m[\/latex] is the slope of the line and\u00a0[latex]b[\/latex] is the [latex]y[\/latex]-intercept.\r\n\r\n<ul>\r\n\t<li>[latex]x[\/latex] is the explanatory (independent) variable.<\/li>\r\n\t<li>[latex]y[\/latex] is the response (dependent) variable, the focus of the study.<\/li>\r\n\t<li>The slope, [latex]m[\/latex], is the ratio that measures how much [latex]y[\/latex] changes compared to an increase in [latex]x[\/latex].\r\n\r\n<ul>\r\n\t<li>[latex] \\displaystyle \\text{Slope }=\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/li>\r\n\t<li>[latex] \\displaystyle m=\\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}[\/latex] where [latex]m=\\text{slope}[\/latex]\u00a0and [latex] \\displaystyle ({{x}_{1}},{{y}_{1}})[\/latex] and [latex] \\displaystyle ({{x}_{2}},{{y}_{2}})[\/latex] are two points on the line.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>The [latex]y[\/latex]-intercept, [latex]b[\/latex], is the place on the graph where the line crosses the [latex]y[\/latex]-axis.<\/li>\r\n<\/ul>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"416\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/09\/23234536\/image031.jpeg\" alt=\"A graph illustrating a straight, diagonal line. Points 1 and 2 are labeled, with how to find the measurements for rise and run detailed.\" width=\"416\" height=\"401\" \/> Figure 1. Graph showing how to calculate the slope of a line using two points: slope is the ratio of the vertical change (rise) to the horizontal change (run).[\/caption] [caption id=\"\" align=\"aligncenter\" width=\"484\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/09\/24000513\/Picture115.png\" alt=\"A graph with another diagonal line, with explanatory variable marking the x axis and the response or predicted value marking the y axis. The y-intercept, b, is marked on the line. m = rise\/run is written in a box, with color-matching lines showing rise and run. \" width=\"484\" height=\"494\" \/> Figure 2. An alternate graph showing how to calculate the slope of a line using two points: slope is the ratio of the vertical change (rise) to the horizontal change (run).[\/caption]\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1165[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1166[\/ohm2_question]<\/section>\r\n<section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1190[\/ohm2_question]<\/section>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Find and explain the slope and y-intercept of a linear model.<\/li>\n<\/ul>\n<\/section>\n<h2>Linear Equation<\/h2>\n<section class=\"textbox recall\">The general form of a linear equation: [latex]y=mx+b[\/latex], where\u00a0[latex]m[\/latex] is the slope of the line and\u00a0[latex]b[\/latex] is the [latex]y[\/latex]-intercept.<\/p>\n<ul>\n<li>[latex]x[\/latex] is the explanatory (independent) variable.<\/li>\n<li>[latex]y[\/latex] is the response (dependent) variable, the focus of the study.<\/li>\n<li>The slope, [latex]m[\/latex], is the ratio that measures how much [latex]y[\/latex] changes compared to an increase in [latex]x[\/latex].\n<ul>\n<li>[latex]\\displaystyle \\text{Slope }=\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/li>\n<li>[latex]\\displaystyle m=\\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}[\/latex] where [latex]m=\\text{slope}[\/latex]\u00a0and [latex]\\displaystyle ({{x}_{1}},{{y}_{1}})[\/latex] and [latex]\\displaystyle ({{x}_{2}},{{y}_{2}})[\/latex] are two points on the line.<\/li>\n<\/ul>\n<\/li>\n<li>The [latex]y[\/latex]-intercept, [latex]b[\/latex], is the place on the graph where the line crosses the [latex]y[\/latex]-axis.<\/li>\n<\/ul>\n<figure style=\"width: 416px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/09\/23234536\/image031.jpeg\" alt=\"A graph illustrating a straight, diagonal line. Points 1 and 2 are labeled, with how to find the measurements for rise and run detailed.\" width=\"416\" height=\"401\" \/><figcaption class=\"wp-caption-text\">Figure 1. Graph showing how to calculate the slope of a line using two points: slope is the ratio of the vertical change (rise) to the horizontal change (run).<\/figcaption><\/figure>\n<figure style=\"width: 484px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/09\/24000513\/Picture115.png\" alt=\"A graph with another diagonal line, with explanatory variable marking the x axis and the response or predicted value marking the y axis. The y-intercept, b, is marked on the line. m = rise\/run is written in a box, with color-matching lines showing rise and run.\" width=\"484\" height=\"494\" \/><figcaption class=\"wp-caption-text\">Figure 2. An alternate graph showing how to calculate the slope of a line using two points: slope is the ratio of the vertical change (rise) to the horizontal change (run).<\/figcaption><\/figure>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1165\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1165&theme=lumen&iframe_resize_id=ohm1165&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1166\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1166&theme=lumen&iframe_resize_id=ohm1166&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1190\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1190&theme=lumen&iframe_resize_id=ohm1190&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n","protected":false},"author":12,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":225,"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\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/254"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":12,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/254\/revisions"}],"predecessor-version":[{"id":6643,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/254\/revisions\/6643"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/225"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/254\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=254"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=254"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=254"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=254"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}