{"id":1312,"date":"2025-07-24T04:14:52","date_gmt":"2025-07-24T04:14:52","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1312"},"modified":"2026-03-09T07:18:26","modified_gmt":"2026-03-09T07:18:26","slug":"linear-functions-fresh-take-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/linear-functions-fresh-take-2\/","title":{"raw":"Linear Functions: Fresh Take","rendered":"Linear Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Write the equation of a linear function given a point and a slope, two points, or a table of values.<\/li>\r\n \t<li>Graph linear functions given any form of its equation.<\/li>\r\n \t<li>Graph and write the equations of horizontal and vertical lines.<\/li>\r\n \t<li>Write the equation of a line parallel or perpendicular to a given line.<\/li>\r\n<\/ul>\r\n<\/section><section id=\"fs-id1165137761836\">\r\n<h2>Writing the Point-Slope Form of a Linear Equation<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea\r\n<\/strong>\r\nPoint-slope form is especially useful when you know the slope and one point on the line, or when you know two points on the line.\r\nThe point-slope form is:\r\n<p style=\"text-align: center;\">[latex]y - y_1 = m(x - x_1)[\/latex]<\/p>\r\nwhere [latex]m[\/latex] is the slope and [latex](x_1, y_1)[\/latex] is a point on the line.\r\n\r\n<\/div>\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Point-slope form comes from the slope formula. Starting with [latex]m = \\frac{y - y_1}{x - x_1}[\/latex], we multiply both sides by [latex](x - x_1)[\/latex] to get [latex]m(x - x_1) = y - y_1[\/latex], which we rearrange as [latex]y - y_1 = m(x - x_1)[\/latex].<\/section>\r\n<h3>Converting Between Forms<\/h3>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n\r\nPoint-slope form and slope-intercept form describe the same line\u2014they're just different ways to write it. We can convert from one form to the other using basic algebra.\r\nFor example, if we have [latex]y - 4 = -\\frac{1}{2}(x - 6)[\/latex] in point-slope form:\r\n[latex]\\begin{align}\r\ny - 4 &amp;= -\\frac{1}{2}(x - 6) \\\\\r\ny - 4 &amp;= -\\frac{1}{2}x + 3 &amp;&amp; \\text{distribute } -\\frac{1}{2} \\\\\r\ny &amp;= -\\frac{1}{2}x + 7 &amp;&amp; \\text{add 4 to both sides}\r\n\\end{align}[\/latex]\r\nBoth equations describe the same line!\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<h3>Writing Equations Using a Point and the Slope<\/h3>\r\n<div class=\"textbox questionHelp\">\r\n\r\n<strong>Question Help: Writing an Equation Given a Point and Slope<\/strong>\r\n<ol>\r\n \t<li>Identify the slope [latex]m[\/latex].<\/li>\r\n \t<li>Identify the coordinates [latex](x_1, y_1)[\/latex] of the point.<\/li>\r\n \t<li>Substitute into point-slope form: [latex]y - y_1 = m(x - x_1)[\/latex].<\/li>\r\n \t<li>If needed, convert to slope-intercept form by distributing and solving for [latex]y[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dacdbdce-FntpEHhLHvw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/FntpEHhLHvw?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dacdbdce-FntpEHhLHvw\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=11328528&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dacdbdce-FntpEHhLHvw&vembed=0&video_id=FntpEHhLHvw&video_target=tpm-plugin-dacdbdce-FntpEHhLHvw'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Writing+an+equation+using+point+slope+form+given+a+point+and+slope_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWriting an equation using point slope form given a point and slope\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h3>Writing Equations Using Two Points<\/h3>\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">When you know two points on a line but don't know the slope, you can find the slope first, then use point-slope form.\r\nThe process:\r\n<ol>\r\n \t<li>Calculate the slope using [latex]m = \\frac{y_2 - y_1}{x_2 - x_1}[\/latex]<\/li>\r\n \t<li>Choose either point to use as [latex](x_1, y_1)[\/latex]<\/li>\r\n \t<li>Substitute into point-slope form<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ccgfecgd-DhDtKR0VyLE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/DhDtKR0VyLE?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ccgfecgd-DhDtKR0VyLE\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660001&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ccgfecgd-DhDtKR0VyLE&vembed=0&video_id=DhDtKR0VyLE&video_target=tpm-plugin-ccgfecgd-DhDtKR0VyLE'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Writing+an+equation+using+point+slope+form+given+two+points_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWriting an equation using point slope form given two points\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<div class=\"textbox proTip\">When you have two points, you can use either one in the point-slope formula\u2014you'll get the same line! The equations might look different in point-slope form, but they'll simplify to the same slope-intercept form.<\/div>\r\n<h3>Writing Equations from a Graph<\/h3>\r\n<div class=\"textbox questionHelp\">\r\n\r\n<strong>Question Help: Writing an Equation from a Graph<\/strong>\r\n<ol>\r\n \t<li>Identify two points on the line.<\/li>\r\n \t<li>Use the two points to calculate the slope.<\/li>\r\n \t<li>Determine where the line crosses the y-axis to identify the y-intercept by visual inspection.<\/li>\r\n \t<li>Substitute the slope and y-intercept into slope-intercept form [latex]y = mx + b[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dbagedbc-L_5tE1vUsyc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/L_5tE1vUsyc?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dbagedbc-L_5tE1vUsyc\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660002&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dbagedbc-L_5tE1vUsyc&vembed=0&video_id=L_5tE1vUsyc&video_target=tpm-plugin-dbagedbc-L_5tE1vUsyc'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Write+a+Slope+Intercept+Equation+for+a+Line+on+a+Graph_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWrite a Slope Intercept Equation for a Line on a Graph\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Modeling Real-World Problems with Linear Functions<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong>\r\nLinear functions model many real-world situations where something changes at a constant rate:The key is identifying:\r\n<ul>\r\n \t<li><strong>Initial value<\/strong> (the y-intercept [latex]b[\/latex]): What you start with<\/li>\r\n \t<li><strong>Rate of change<\/strong> (the slope [latex]m[\/latex]): How much changes per unit of time\/input<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">Marcus currently has 200 songs in his music collection. Every month, he adds 15 new songs. Write a formula for the number of songs, [latex]N[\/latex], in his collection as a function of time, [latex]t[\/latex] (in months). How many songs will he have in a year?\r\n[reveal-answer q=\"lin3\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"lin3\"]\r\n<strong>Identify the parts:<\/strong>\r\n<ul>\r\n \t<li>Initial value: [latex]N(0) = 200[\/latex], so [latex]b = 200[\/latex]<\/li>\r\n \t<li>Rate of change: 15 songs per month, so [latex]m = 15[\/latex]<\/li>\r\n<\/ul>\r\n<strong>Write the equation:<\/strong>\r\n[latex]N(t) = 15t + 200[\/latex]\r\n<strong>Find songs after 1 year (12 months):<\/strong>\r\n[latex]\\begin{align}\r\nN(12) &amp;= 15(12) + 200 \\\\\r\n&amp;= 180 + 200 \\\\\r\n&amp;= 380\r\n\\end{align}[\/latex]\r\nMarcus will have 380 songs in 12 months.\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">Working as an insurance salesperson, Ilya earns a base salary plus commission on each new policy. Last week he sold 3 policies and earned $760. The week before, he sold 5 policies and earned $920. Find an equation for [latex]I(n)[\/latex], his weekly income as a function of policies sold, and interpret its meaning.\r\n[reveal-answer q=\"lin4\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"lin4\"]\r\nWe have two input-output pairs: [latex](3, 760)[\/latex] and [latex](5, 920)[\/latex].\r\n<strong>\r\nFind the rate of change (commission per policy):<\/strong>\r\n[latex]\\begin{align}\r\nm &amp;= \\frac{920 - 760}{5 - 3} \\\\\r\n&amp;= \\frac{160}{2} \\\\\r\n&amp;= 80 \\text{ dollars per policy}\r\n\\end{align}[\/latex]\r\n<strong>\r\nFind the initial value (base salary):<\/strong>\r\n[latex]\\begin{align}\r\nI(n) &amp;= 80n + b \\\\\r\n760 &amp;= 80(3) + b &amp;&amp; \\text{substitute known point} \\\\\r\n760 &amp;= 240 + b \\\\\r\n520 &amp;= b\r\n\\end{align}[\/latex]\r\n<strong>\r\nWrite the equation:<\/strong>\r\n[latex]I(n) = 80n + 520[\/latex]\r\n<strong>\r\nInterpretation:<\/strong> Ilya's base salary is $520 per week, and he earns an additional $80 commission for each policy sold.\r\n[\/hidden-answer]<\/section>\r\n<div class=\"textbox recall\">In linear models, the slope represents the rate of change (how much something increases per unit), and the y-intercept represents the starting value or fixed amount.<\/div>\r\n<h2>Horizontal and Vertical Lines<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n\r\nHorizontal lines have a slope of 0. The y-value is constant for all x-values. Equation form: [latex]y = c[\/latex] (where [latex]c[\/latex] is a constant).\r\n\r\nVertical lines have an undefined slope. The x-value is constant for all y-values. Equation form: [latex]x = a[\/latex] (where [latex]a[\/latex] is a constant).\r\n\r\n<\/div>\r\n<div class=\"textbox proTip\">A horizontal line is a function (it passes the vertical line test), but a vertical line is NOT a function (it fails the vertical line test because one input maps to infinitely many outputs).<\/div>\r\n<\/section>\r\n<div><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dbcggahb-G8epj0-fw1Q\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/G8epj0-fw1Q?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dbcggahb-G8epj0-fw1Q\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660003&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dbcggahb-G8epj0-fw1Q&vembed=0&video_id=G8epj0-fw1Q&video_target=tpm-plugin-dbcggahb-G8epj0-fw1Q'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Horizontal+and+Vertical+Lines+(How+to+Graph+and+Write+Equations)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHorizontal and Vertical Lines (How to Graph and Write Equations)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<section id=\"fs-id1165137761836\">\r\n<h2>Parallel and Perpendicular Lines<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong>\r\n<strong>\r\n<\/strong>Parallel lines never intersect. They have the same slope but different y-intercepts.Perpendicular lines intersect at right angles (90\u00b0). Their slopes are negative reciprocals of each other.<\/div>\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bfhbagce-errqXpHe510\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/errqXpHe510?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bfhbagce-errqXpHe510\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660004&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bfhbagce-errqXpHe510&vembed=0&video_id=errqXpHe510&video_target=tpm-plugin-bfhbagce-errqXpHe510'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Determine+if+Lines+are+Parallel%2C+Perpendicular+or+Neither_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermine if Lines are Parallel, Perpendicular or Neither\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h3>Writing Equations of Parallel Lines<\/h3>\r\n<div class=\"textbox questionHelp\">\r\n\r\n<strong>Question Help: Writing a Parallel Line Equation<\/strong>\r\n<ol>\r\n \t<li>Find the slope of the given function.<\/li>\r\n \t<li>Use the same slope for the parallel line.<\/li>\r\n \t<li>Substitute the slope and the given point into either point-slope form or slope-intercept form.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ddbbedhh-0iQI99ov34I\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/0iQI99ov34I?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ddbbedhh-0iQI99ov34I\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660005&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ddbbedhh-0iQI99ov34I&vembed=0&video_id=0iQI99ov34I&video_target=tpm-plugin-ddbbedhh-0iQI99ov34I'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Writing+Equations+of+Parallel+Lines+Tutorial_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWriting Equations of Parallel Lines Tutorial\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h3>Writing Equations of Perpendicular Lines<\/h3>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong>\r\nTo write an equation of a line perpendicular to a given line:\r\n<ol>\r\n \t<li>Find the slope of the given line<\/li>\r\n \t<li>Find the negative reciprocal of that slope<\/li>\r\n \t<li>Use this new slope and the given point to write the equation<\/li>\r\n \t<li>Simplify if needed<\/li>\r\n<\/ol>\r\nTo find the negative reciprocal: flip the fraction and change the sign.\r\n<ul>\r\n \t<li>Slope 2 \u2192 negative reciprocal is [latex]-\\frac{1}{2}[\/latex]<\/li>\r\n \t<li>Slope [latex]-\\frac{3}{4}[\/latex] \u2192 negative reciprocal is [latex]\\frac{4}{3}[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox questionHelp\">\r\n\r\n<strong>Question Help: Writing a Perpendicular Line Equation<\/strong>\r\n<ol>\r\n \t<li>Find the slope of the given function.<\/li>\r\n \t<li>Determine the negative reciprocal of the slope.<\/li>\r\n \t<li>Substitute the new slope and the values for [latex]x[\/latex] and [latex]y[\/latex] from the given point into [latex]y = mx + b[\/latex].<\/li>\r\n \t<li>Solve for [latex]b[\/latex].<\/li>\r\n \t<li>Write the equation for the line.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ffgbeehh-3ewIVsnHjeA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/3ewIVsnHjeA?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ffgbeehh-3ewIVsnHjeA\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660006&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ffgbeehh-3ewIVsnHjeA&vembed=0&video_id=3ewIVsnHjeA&video_target=tpm-plugin-ffgbeehh-3ewIVsnHjeA'><\/script><\/p>\r\nYou can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Writing+Equations+of+Perpendicular+Lines+Tutorial_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWriting Equations of Perpendicular Lines Tutorial\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Write the equation of a linear function given a point and a slope, two points, or a table of values.<\/li>\n<li>Graph linear functions given any form of its equation.<\/li>\n<li>Graph and write the equations of horizontal and vertical lines.<\/li>\n<li>Write the equation of a line parallel or perpendicular to a given line.<\/li>\n<\/ul>\n<\/section>\n<section id=\"fs-id1165137761836\">\n<h2>Writing the Point-Slope Form of a Linear Equation<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<br \/>\n<\/strong><br \/>\nPoint-slope form is especially useful when you know the slope and one point on the line, or when you know two points on the line.<br \/>\nThe point-slope form is:<\/p>\n<p style=\"text-align: center;\">[latex]y - y_1 = m(x - x_1)[\/latex]<\/p>\n<p>where [latex]m[\/latex] is the slope and [latex](x_1, y_1)[\/latex] is a point on the line.<\/p>\n<\/div>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Point-slope form comes from the slope formula. Starting with [latex]m = \\frac{y - y_1}{x - x_1}[\/latex], we multiply both sides by [latex](x - x_1)[\/latex] to get [latex]m(x - x_1) = y - y_1[\/latex], which we rearrange as [latex]y - y_1 = m(x - x_1)[\/latex].<\/section>\n<h3>Converting Between Forms<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<p>Point-slope form and slope-intercept form describe the same line\u2014they&#8217;re just different ways to write it. We can convert from one form to the other using basic algebra.<br \/>\nFor example, if we have [latex]y - 4 = -\\frac{1}{2}(x - 6)[\/latex] in point-slope form:<br \/>\n[latex]\\begin{align}  y - 4 &= -\\frac{1}{2}(x - 6) \\\\  y - 4 &= -\\frac{1}{2}x + 3 && \\text{distribute } -\\frac{1}{2} \\\\  y &= -\\frac{1}{2}x + 7 && \\text{add 4 to both sides}  \\end{align}[\/latex]<br \/>\nBoth equations describe the same line!<\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<\/div>\n<h3>Writing Equations Using a Point and the Slope<\/h3>\n<div class=\"textbox questionHelp\">\n<p><strong>Question Help: Writing an Equation Given a Point and Slope<\/strong><\/p>\n<ol>\n<li>Identify the slope [latex]m[\/latex].<\/li>\n<li>Identify the coordinates [latex](x_1, y_1)[\/latex] of the point.<\/li>\n<li>Substitute into point-slope form: [latex]y - y_1 = m(x - x_1)[\/latex].<\/li>\n<li>If needed, convert to slope-intercept form by distributing and solving for [latex]y[\/latex].<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dacdbdce-FntpEHhLHvw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/FntpEHhLHvw?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dacdbdce-FntpEHhLHvw\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=11328528&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-dacdbdce-FntpEHhLHvw&#38;vembed=0&#38;video_id=FntpEHhLHvw&#38;video_target=tpm-plugin-dacdbdce-FntpEHhLHvw\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Writing+an+equation+using+point+slope+form+given+a+point+and+slope_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWriting an equation using point slope form given a point and slope\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Writing Equations Using Two Points<\/h3>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">When you know two points on a line but don&#8217;t know the slope, you can find the slope first, then use point-slope form.<br \/>\nThe process:<\/p>\n<ol>\n<li>Calculate the slope using [latex]m = \\frac{y_2 - y_1}{x_2 - x_1}[\/latex]<\/li>\n<li>Choose either point to use as [latex](x_1, y_1)[\/latex]<\/li>\n<li>Substitute into point-slope form<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ccgfecgd-DhDtKR0VyLE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/DhDtKR0VyLE?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ccgfecgd-DhDtKR0VyLE\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14660001&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ccgfecgd-DhDtKR0VyLE&#38;vembed=0&#38;video_id=DhDtKR0VyLE&#38;video_target=tpm-plugin-ccgfecgd-DhDtKR0VyLE\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Writing+an+equation+using+point+slope+form+given+two+points_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWriting an equation using point slope form given two points\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<div class=\"textbox proTip\">When you have two points, you can use either one in the point-slope formula\u2014you&#8217;ll get the same line! The equations might look different in point-slope form, but they&#8217;ll simplify to the same slope-intercept form.<\/div>\n<h3>Writing Equations from a Graph<\/h3>\n<div class=\"textbox questionHelp\">\n<p><strong>Question Help: Writing an Equation from a Graph<\/strong><\/p>\n<ol>\n<li>Identify two points on the line.<\/li>\n<li>Use the two points to calculate the slope.<\/li>\n<li>Determine where the line crosses the y-axis to identify the y-intercept by visual inspection.<\/li>\n<li>Substitute the slope and y-intercept into slope-intercept form [latex]y = mx + b[\/latex].<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dbagedbc-L_5tE1vUsyc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/L_5tE1vUsyc?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dbagedbc-L_5tE1vUsyc\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14660002&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-dbagedbc-L_5tE1vUsyc&#38;vembed=0&#38;video_id=L_5tE1vUsyc&#38;video_target=tpm-plugin-dbagedbc-L_5tE1vUsyc\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Write+a+Slope+Intercept+Equation+for+a+Line+on+a+Graph_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWrite a Slope Intercept Equation for a Line on a Graph\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Modeling Real-World Problems with Linear Functions<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong><br \/>\nLinear functions model many real-world situations where something changes at a constant rate:The key is identifying:<\/p>\n<ul>\n<li><strong>Initial value<\/strong> (the y-intercept [latex]b[\/latex]): What you start with<\/li>\n<li><strong>Rate of change<\/strong> (the slope [latex]m[\/latex]): How much changes per unit of time\/input<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">Marcus currently has 200 songs in his music collection. Every month, he adds 15 new songs. Write a formula for the number of songs, [latex]N[\/latex], in his collection as a function of time, [latex]t[\/latex] (in months). How many songs will he have in a year?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qlin3\">Show Solution<\/button><\/p>\n<div id=\"qlin3\" class=\"hidden-answer\" style=\"display: none\">\n<strong>Identify the parts:<\/strong><\/p>\n<ul>\n<li>Initial value: [latex]N(0) = 200[\/latex], so [latex]b = 200[\/latex]<\/li>\n<li>Rate of change: 15 songs per month, so [latex]m = 15[\/latex]<\/li>\n<\/ul>\n<p><strong>Write the equation:<\/strong><br \/>\n[latex]N(t) = 15t + 200[\/latex]<br \/>\n<strong>Find songs after 1 year (12 months):<\/strong><br \/>\n[latex]\\begin{align}  N(12) &= 15(12) + 200 \\\\  &= 180 + 200 \\\\  &= 380  \\end{align}[\/latex]<br \/>\nMarcus will have 380 songs in 12 months.\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Working as an insurance salesperson, Ilya earns a base salary plus commission on each new policy. Last week he sold 3 policies and earned $760. The week before, he sold 5 policies and earned $920. Find an equation for [latex]I(n)[\/latex], his weekly income as a function of policies sold, and interpret its meaning.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qlin4\">Show Solution<\/button><\/p>\n<div id=\"qlin4\" class=\"hidden-answer\" style=\"display: none\">\nWe have two input-output pairs: [latex](3, 760)[\/latex] and [latex](5, 920)[\/latex].<br \/>\n<strong><br \/>\nFind the rate of change (commission per policy):<\/strong><br \/>\n[latex]\\begin{align}  m &= \\frac{920 - 760}{5 - 3} \\\\  &= \\frac{160}{2} \\\\  &= 80 \\text{ dollars per policy}  \\end{align}[\/latex]<br \/>\n<strong><br \/>\nFind the initial value (base salary):<\/strong><br \/>\n[latex]\\begin{align}  I(n) &= 80n + b \\\\  760 &= 80(3) + b && \\text{substitute known point} \\\\  760 &= 240 + b \\\\  520 &= b  \\end{align}[\/latex]<br \/>\n<strong><br \/>\nWrite the equation:<\/strong><br \/>\n[latex]I(n) = 80n + 520[\/latex]<br \/>\n<strong><br \/>\nInterpretation:<\/strong> Ilya&#8217;s base salary is $520 per week, and he earns an additional $80 commission for each policy sold.\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox recall\">In linear models, the slope represents the rate of change (how much something increases per unit), and the y-intercept represents the starting value or fixed amount.<\/div>\n<h2>Horizontal and Vertical Lines<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<p>Horizontal lines have a slope of 0. The y-value is constant for all x-values. Equation form: [latex]y = c[\/latex] (where [latex]c[\/latex] is a constant).<\/p>\n<p>Vertical lines have an undefined slope. The x-value is constant for all y-values. Equation form: [latex]x = a[\/latex] (where [latex]a[\/latex] is a constant).<\/p>\n<\/div>\n<div class=\"textbox proTip\">A horizontal line is a function (it passes the vertical line test), but a vertical line is NOT a function (it fails the vertical line test because one input maps to infinitely many outputs).<\/div>\n<\/section>\n<div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dbcggahb-G8epj0-fw1Q\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/G8epj0-fw1Q?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dbcggahb-G8epj0-fw1Q\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14660003&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-dbcggahb-G8epj0-fw1Q&#38;vembed=0&#38;video_id=G8epj0-fw1Q&#38;video_target=tpm-plugin-dbcggahb-G8epj0-fw1Q\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Horizontal+and+Vertical+Lines+(How+to+Graph+and+Write+Equations)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHorizontal and Vertical Lines (How to Graph and Write Equations)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<section>\n<h2>Parallel and Perpendicular Lines<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong><br \/>\n<strong><br \/>\n<\/strong>Parallel lines never intersect. They have the same slope but different y-intercepts.Perpendicular lines intersect at right angles (90\u00b0). Their slopes are negative reciprocals of each other.<\/div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bfhbagce-errqXpHe510\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/errqXpHe510?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bfhbagce-errqXpHe510\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14660004&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-bfhbagce-errqXpHe510&#38;vembed=0&#38;video_id=errqXpHe510&#38;video_target=tpm-plugin-bfhbagce-errqXpHe510\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Determine+if+Lines+are+Parallel%2C+Perpendicular+or+Neither_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermine if Lines are Parallel, Perpendicular or Neither\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Writing Equations of Parallel Lines<\/h3>\n<div class=\"textbox questionHelp\">\n<p><strong>Question Help: Writing a Parallel Line Equation<\/strong><\/p>\n<ol>\n<li>Find the slope of the given function.<\/li>\n<li>Use the same slope for the parallel line.<\/li>\n<li>Substitute the slope and the given point into either point-slope form or slope-intercept form.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ddbbedhh-0iQI99ov34I\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/0iQI99ov34I?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ddbbedhh-0iQI99ov34I\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14660005&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ddbbedhh-0iQI99ov34I&#38;vembed=0&#38;video_id=0iQI99ov34I&#38;video_target=tpm-plugin-ddbbedhh-0iQI99ov34I\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Writing+Equations+of+Parallel+Lines+Tutorial_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWriting Equations of Parallel Lines Tutorial\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Writing Equations of Perpendicular Lines<\/h3>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong><br \/>\nTo write an equation of a line perpendicular to a given line:<\/p>\n<ol>\n<li>Find the slope of the given line<\/li>\n<li>Find the negative reciprocal of that slope<\/li>\n<li>Use this new slope and the given point to write the equation<\/li>\n<li>Simplify if needed<\/li>\n<\/ol>\n<p>To find the negative reciprocal: flip the fraction and change the sign.<\/p>\n<ul>\n<li>Slope 2 \u2192 negative reciprocal is [latex]-\\frac{1}{2}[\/latex]<\/li>\n<li>Slope [latex]-\\frac{3}{4}[\/latex] \u2192 negative reciprocal is [latex]\\frac{4}{3}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox questionHelp\">\n<p><strong>Question Help: Writing a Perpendicular Line Equation<\/strong><\/p>\n<ol>\n<li>Find the slope of the given function.<\/li>\n<li>Determine the negative reciprocal of the slope.<\/li>\n<li>Substitute the new slope and the values for [latex]x[\/latex] and [latex]y[\/latex] from the given point into [latex]y = mx + b[\/latex].<\/li>\n<li>Solve for [latex]b[\/latex].<\/li>\n<li>Write the equation for the line.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ffgbeehh-3ewIVsnHjeA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/3ewIVsnHjeA?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ffgbeehh-3ewIVsnHjeA\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14660006&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ffgbeehh-3ewIVsnHjeA&#38;vembed=0&#38;video_id=3ewIVsnHjeA&#38;video_target=tpm-plugin-ffgbeehh-3ewIVsnHjeA\"><\/script><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Writing+Equations+of+Perpendicular+Lines+Tutorial_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWriting Equations of Perpendicular Lines Tutorial\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/section>\n","protected":false},"author":67,"menu_order":22,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Writing an equation using point slope form given a point and slope\",\"author\":\"Brian McLogan\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/FntpEHhLHvw\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Writing an equation using point slope form given two points\",\"author\":\"Brian McLogan\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/DhDtKR0VyLE\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Write a Slope Intercept Equation for a Line on a Graph\",\"author\":\"\",\"organization\":\"Davitily\",\"url\":\"https:\/\/youtu.be\/L_5tE1vUsyc\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Horizontal and Vertical Lines (How to Graph and Write Equations)\",\"author\":\"\",\"organization\":\"Mario\\'s Math Tutoring\",\"url\":\"https:\/\/youtu.be\/G8epj0-fw1Q\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Determine if Lines are Parallel, Perpendicular or Neither\",\"author\":\"\",\"organization\":\"Professor Kat\",\"url\":\"https:\/\/youtu.be\/errqXpHe510\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Writing Equations of Parallel Lines Tutorial\",\"author\":\"\",\"organization\":\"Friendly Math 101\",\"url\":\"https:\/\/youtu.be\/0iQI99ov34I\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Writing Equations of Perpendicular Lines Tutorial\",\"author\":\"\",\"organization\":\"Friendly Math 101\",\"url\":\"https:\/\/youtu.be\/3ewIVsnHjeA\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":61,"module-header":"fresh_take","content_attributions":null,"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=11328528&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dacdbdce-FntpEHhLHvw&vembed=0&video_id=FntpEHhLHvw&video_target=tpm-plugin-dacdbdce-FntpEHhLHvw'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660001&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ccgfecgd-DhDtKR0VyLE&vembed=0&video_id=DhDtKR0VyLE&video_target=tpm-plugin-ccgfecgd-DhDtKR0VyLE'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660002&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dbagedbc-L_5tE1vUsyc&vembed=0&video_id=L_5tE1vUsyc&video_target=tpm-plugin-dbagedbc-L_5tE1vUsyc'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660003&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dbcggahb-G8epj0-fw1Q&vembed=0&video_id=G8epj0-fw1Q&video_target=tpm-plugin-dbcggahb-G8epj0-fw1Q'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660004&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bfhbagce-errqXpHe510&vembed=0&video_id=errqXpHe510&video_target=tpm-plugin-bfhbagce-errqXpHe510'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660005&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ddbbedhh-0iQI99ov34I&vembed=0&video_id=0iQI99ov34I&video_target=tpm-plugin-ddbbedhh-0iQI99ov34I'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660006&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ffgbeehh-3ewIVsnHjeA&vembed=0&video_id=3ewIVsnHjeA&video_target=tpm-plugin-ffgbeehh-3ewIVsnHjeA'><\/script>\n","media_targets":["tpm-plugin-dacdbdce-FntpEHhLHvw","tpm-plugin-ccgfecgd-DhDtKR0VyLE","tpm-plugin-dbagedbc-L_5tE1vUsyc","tpm-plugin-dbcggahb-G8epj0-fw1Q","tpm-plugin-bfhbagce-errqXpHe510","tpm-plugin-ddbbedhh-0iQI99ov34I","tpm-plugin-ffgbeehh-3ewIVsnHjeA"]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1312"}],"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":9,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1312\/revisions"}],"predecessor-version":[{"id":5747,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1312\/revisions\/5747"}],"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\/1312\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1312"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1312"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1312"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1312"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}