{"id":1063,"date":"2024-05-02T22:51:26","date_gmt":"2024-05-02T22:51:26","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1063"},"modified":"2025-09-22T22:06:46","modified_gmt":"2025-09-22T22:06:46","slug":"equations-of-lines-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/equations-of-lines-fresh-take\/","title":{"raw":"Equations of Lines: Fresh Take","rendered":"Equations of Lines: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Find the value of a variable that satisfies an equation<\/li>\r\n \t<li>Write equations for lines using different forms: slope-intercept, point-slope, and standard form<\/li>\r\n \t<li>Recognize and write equations for horizontal and vertical lines<\/li>\r\n \t<li>Determine if lines are parallel or perpendicular, and write equations for lines parallel or perpendicular to a given line<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Solving Linear Equations in One Variable<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Equation Basics:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Contains an equal sign (=) separating two expressions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">May include coefficients, variables, terms, and expressions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Goal: Isolate the variable on one side<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Types of Equations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Simple equations (solved mentally): [latex]2x = 6[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multi-step equations (require multiple operations): [latex]4(\\frac{1}{3}t + \\frac{1}{2}) = 6[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Equations with variables on both sides: [latex]4x - 6 = 2x + 10[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Properties:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Addition Property of Equality: If [latex]a = b[\/latex], then [latex]a + c = b + c[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiplication Property of Equality: If [latex]a = b[\/latex], then [latex]ac = bc[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">These properties maintain the balance of the equation<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solving Strategy:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Simplify expressions on both sides<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Isolate variable terms on one side<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Isolate the variable itself<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Common Techniques:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Combining like terms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Clearing fractions or decimals<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Moving terms between sides of the equation<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Solving Process<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Simplify each side of the equation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Clear parentheses<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Combine like terms<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Move variable terms to one side:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Add or subtract to move terms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Choose the side with the larger coefficient (if applicable)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Move constant terms to the other side:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Add or subtract as needed<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Isolate the variable:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Multiply or divide both sides<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check the solution:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Substitute the result back into the original equation<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Solve the equation: [latex]2(3x - 4) - 5 = 4x + 7[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[reveal-answer q=\"610206\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"610206\"]<\/p>\r\n\r\n<ol>\r\n \t<li class=\"whitespace-normal break-words\">Simplify the left side by distributing: [latex]2(3x - 4) - 5 = 4x + 7[\/latex] [latex]6x - 8 - 5 = 4x + 7[\/latex] [latex]6x - 13 = 4x + 7[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Move variable terms to one side (left) using the Addition Property of Equality: [latex]6x - 4x = 7 + 13[\/latex] [latex]2x = 20[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Isolate the variable using the Multiplication Property of Equality: [latex]x = \\frac{20}{2} = 10[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check the solution: Left side: [latex]2(3(10) - 4) - 5 = 2(30 - 4) - 5 = 2(26) - 5 = 52 - 5 = 47[\/latex] Right side: [latex]4(10) + 7 = 40 + 7 = 47[\/latex] The solution [latex]x = 10[\/latex] satisfies the original equation.<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">[\/hidden-answer]<\/p>\r\n\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-effhggcc-Wj2WCzzEL28\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Wj2WCzzEL28?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-effhggcc-Wj2WCzzEL28\" 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=13881110&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-effhggcc-Wj2WCzzEL28&vembed=0&video_id=Wj2WCzzEL28&video_target=tpm-plugin-effhggcc-Wj2WCzzEL28'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/How+to+solve+Linear+Equations+with+variables+on+both+sides_+Linear+Algebra+Education_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Solve Linear Equations With Variables on Both Sides : Linear Algebra Education\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Linear Equations<\/h2>\r\n<h3>Slope-Intercept Form<\/h3>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Equation form: [latex]y = mx + b[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]m[\/latex] represents the slope<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]b[\/latex] represents the y-intercept<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Slope ([latex]m[\/latex]):\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Measures the steepness of the line<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculated as [latex]m = \\frac{\\text{rise}}{\\text{run}} = \\frac{y_2 - y_1}{x_2 - x_1}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Positive slope: line rises from left to right<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Negative slope: line falls from left to right<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Zero slope: horizontal line<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Undefined slope: vertical line<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Y-intercept ([latex]b[\/latex]):\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Point where the line crosses the y-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Occurs when [latex]x = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Represents the starting point of the line on the y-axis<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Interpreting the equation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]m[\/latex] tells how much [latex]y[\/latex] changes for each unit increase in [latex]x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]b[\/latex] tells the value of [latex]y[\/latex] when [latex]x = 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Given [latex]m=4[\/latex], find the equation of the line in slope-intercept form passing through the point [latex]\\left(2,5\\right)[\/latex].[reveal-answer q=\"634647\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"634647\"][latex]y=4x - 3[\/latex][\/hidden-answer]<\/section>\r\n<h3>The Point-Slope Formula<\/h3>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Equation form: [latex]y - y_1 = m(x - x_1)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]m[\/latex] is the slope of the line<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex](x_1, y_1)[\/latex] is a known point on the line<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Components:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Slope ([latex]m[\/latex]): Rate of change of the line<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Point [latex](x_1, y_1)[\/latex]: Any known point on the line<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Usage:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Useful when given a point and slope<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Can be easily converted to slope-intercept form<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Often used in calculus for tangent line equations<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Flexibility:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Any point on the line can be used as [latex](x_1, y_1)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Allows for easy verification of points on the line<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the equation of a line with slope [latex]m = -2[\/latex] passing through the point [latex](3, 5)[\/latex]. Give your answer in point-slope form.[reveal-answer q=\"492478\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"492478\"]\r\n<ol>\r\n \t<li class=\"whitespace-normal break-words\">Identify the components:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Slope: [latex]m = -2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Point: [latex](x_1, y_1) = (3, 5)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute into the point-slope formula: [latex]y - y_1 = m(x - x_1)[\/latex] [latex]y - 5 = -2(x - 3)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">This is the equation in point-slope form. To convert to slope-intercept form: [latex] \\begin{array}{rcl} y - 5 &amp;=&amp; -2(x - 3) \\[2ex] y - 5 &amp;=&amp; -2x + 6 \\[2ex] y &amp;=&amp; -2x + 11 \\end{array} [\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\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-acdcgcgb-SemcMTLjSiw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/SemcMTLjSiw?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-acdcgcgb-SemcMTLjSiw\" 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=12779085&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-acdcgcgb-SemcMTLjSiw&vembed=0&video_id=SemcMTLjSiw&video_target=tpm-plugin-acdcgcgb-SemcMTLjSiw'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Finding+the+Equation+of+a+Line+Given+Slope+and+a+Point+-+Point+-+Slope+Form_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFinding the Equation of a Line Given Slope and a Point - Point - Slope Form\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h3>Standard Form of a Line<\/h3>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Equation form: [latex]Ax + By = C[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]A[\/latex], [latex]B[\/latex], and [latex]C[\/latex] are integers<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]A[\/latex] and [latex]B[\/latex] are not both zero<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Components:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]A[\/latex]: Coefficient of x<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]B[\/latex]: Coefficient of y<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]C[\/latex]: Constant term<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Characteristics:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">All terms are on one side of the equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">x and y terms are on the left, constant on the right<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Coefficients are typically reduced to lowest terms<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Conversions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Can be converted from\/to slope-intercept form<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Can be converted from\/to point-slope form<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the equation of the line in standard form with slope [latex]m=-\\frac{1}{3}[\/latex] which passes through the point [latex]\\left(1,\\frac{1}{3}\\right)[\/latex].[reveal-answer q=\"3712\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"3712\"][latex]x+3y=2[\/latex][\/hidden-answer]<\/section><section aria-label=\"Example\"><\/section><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-hgadahda-q88S98Y_Pp0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/q88S98Y_Pp0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hgadahda-q88S98Y_Pp0\" 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=12779135&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hgadahda-q88S98Y_Pp0&vembed=0&video_id=q88S98Y_Pp0&video_target=tpm-plugin-hgadahda-q88S98Y_Pp0'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Write+Standard+Form+(when+given+point+and+slope)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWrite Standard Form (when given point and slope)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/section><section class=\"textbox example\">Find the equation of the line that passes through [latex](1,2)[\/latex] and [latex](3,10)[\/latex] in slope-intercept, point-slope, and standard form.[latex]\\begin{align*} \\text{Given Points: } &amp; (1, 2) \\text{ and } (3, 10)\\\\ \\text{Calculate Slope: } &amp; \\\\ &amp; m = \\frac{10 - 2}{3 - 1} = \\frac{8}{2} = 4 \\\\ \\text{Point-Slope Form: } &amp; \\\\ &amp; y - 2 = 4(x - 1) \\\\ \\text{Expand to Slope-Intercept Form: } &amp; \\\\ &amp; y - 2 = 4x - 4 \\\\ &amp; y = 4x - 2 \\\\ \\text{Convert to Standard Form: } &amp; \\\\ &amp; -4x + y = -2 \\\\ &amp; 4x - y = 2 \\quad \\text{(Make \\(A\\) positive)} \\end{align*}[\/latex]<\/section>\r\n<h2>Vertical and Horizontal Lines<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Vertical Lines:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Equation: [latex]x = c[\/latex], where [latex]c[\/latex] is a constant<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Slope: Undefined<\/li>\r\n \t<li class=\"whitespace-normal break-words\">All points have the same x-coordinate<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Cannot be expressed in slope-intercept form<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Represent a constant [latex]x[\/latex]-value for all [latex]y[\/latex]-values<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal Lines:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Equation: [latex]y = c[\/latex], where [latex]c[\/latex] is a constant<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Slope: Zero<\/li>\r\n \t<li class=\"whitespace-normal break-words\">All points have the same [latex]y[\/latex]-coordinate<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Can be expressed in slope-intercept form with [latex]m = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Represent a constant [latex]y[\/latex]-value for all [latex]x[\/latex]-values<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Characteristics:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Vertical lines are parallel to the y-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal lines are parallel to the x-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">These lines are perpendicular to each other<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Identifying Line Type:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li>If all points share the same [latex]x[\/latex]-coordinate: Vertical line<\/li>\r\n \t<li>If all points share the same [latex]y[\/latex]-coordinate: Horizontal line<\/li>\r\n \t<li>If neither: Oblique (slanted) line<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the equation of the line passing through the given points: [latex]\\left(1,-3\\right)[\/latex] and [latex]\\left(1,4\\right)[\/latex].\r\n[reveal-answer q=\"122244\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"122244\"]The [latex]x[\/latex]<em>-<\/em>coordinate of both points is [latex]1[\/latex]. Therefore, we have a vertical line, [latex]x=1[\/latex].[\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the equation of the line passing through [latex]\\left(-5,2\\right)[\/latex] and [latex]\\left(2,2\\right)[\/latex].[reveal-answer q=\"864962\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"864962\"]Horizontal line: [latex]y=2[\/latex][\/hidden-answer]<\/section>\r\n<h2>Parallel and Perpendicular Lines<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li>Parallel Lines:\r\n<ul>\r\n \t<li>Equation: Lines have the same slope but different y-intercepts.<\/li>\r\n \t<li>Slope: Same for all parallel lines.<\/li>\r\n \t<li>Properties:\r\n<ul>\r\n \t<li>Parallel lines never intersect.<\/li>\r\n \t<li>The distance between the lines is constant (equidistant).<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Perpendicular Lines:\r\n<ul>\r\n \t<li>Equation: Lines have slopes that are negative reciprocals of each other.<\/li>\r\n \t<li>Slope: The product of the slopes of two perpendicular lines is -1.<\/li>\r\n \t<li>Properties:\r\n<ul>\r\n \t<li>Perpendicular lines intersect at a 90-degree angle.<\/li>\r\n \t<li>If one line has slope [latex]m[\/latex], the perpendicular line will have slope [latex]-\\dfrac{1}{m}[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Identifying Line Type:\r\n<ul>\r\n \t<li>Parallel Lines: Look for lines with identical slopes and different y-intercepts.<\/li>\r\n \t<li>Perpendicular Lines: Check if the product of their slopes is -1.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Writing equations of parallel and perpendicular lines\r\n<ul>\r\n \t<li>To write the equation of a line <strong>parallel<\/strong> to a given line, use the same slope as the original line but adjust the y-intercept based on a point the line passes through.<\/li>\r\n \t<li>To write the equation of a line <strong>perpendicular<\/strong> to a given line, use the negative reciprocal of the original slope and find the y-intercept using a point the line passes through.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">Graph the two lines and determine whether they are parallel, perpendicular, or neither: [latex]2y-x=10[\/latex] and [latex]2y=x+4[\/latex].[reveal-answer q=\"727314\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"727314\"]Parallel lines. Write the equations in slope-intercept form.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200328\/CNX_CAT_Figure_02_02_007.jpg\" alt=\"Coordinate plane with the x-axis ranging from negative 5 to 5 and the y-axis ranging from negative 1 to 6. Two functions are graphed on the same plot: y = x\/2 plus 5 and y = x\/2 plus 2. The lines do not cross.\" width=\"487\" height=\"329\" \/> Graph of y = 1\/2x + 5 and y = 1\/2x + 2[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">A line passes through the points [latex](\u20132, 6)[\/latex] and [latex](4, 5)[\/latex]. Find the equation of a perpendicular line that passes through the point [latex](4, 5)[\/latex].[reveal-answer q=\"119051\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"119051\"]\r\nFrom the two points of the given line, we can calculate the slope of that line.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}{m}_{1}=\\frac{5 - 6}{4-\\left(-2\\right)}\\hfill &amp; =\\frac{-1}{6}\\hfill &amp; =-\\frac{1}{6}\\hfill \\end{array}[\/latex]<\/p>\r\nFind the negative reciprocal of the slope.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}{m}_{2}=\\frac{-1}{-\\frac{1}{6}}\\hfill &amp; =-1\\left(-\\frac{6}{1}\\right)\\hfill &amp; =6\\hfill \\end{array}[\/latex]<\/p>\r\nWe can then solve for the <em>y-<\/em>intercept of the line passing through the point (4, 5).\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lllll}y=6x+b\\hfill \\\\ 5=6\\left(4\\right)+b\\hfill \\\\ 5=24+b\\hfill \\\\ -19=b\\hfill \\\\ b=-19\\hfill \\end{array}[\/latex]<\/p>\r\nThe equation for the line that is perpendicular to the line passing through the two given points and also passes through point (4, 5) is:\r\n<p style=\"text-align: center;\">[latex]y=6x - 19[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">Write the equation of line parallel to a [latex]5x+3y=1[\/latex] which passes through the point [latex]\\left(3,5\\right)[\/latex].\r\n[reveal-answer q=\"72460\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"72460\"]First, we will write the equation in slope-intercept form to find the slope.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}5x+3y=1\\hfill \\\\ 3y=-5x+1\\hfill \\\\ y=-\\frac{5}{3}x+\\frac{1}{3}\\hfill \\end{array}[\/latex]<\/p>\r\nThe slope is [latex]m=-\\frac{5}{3}[\/latex]. The <em>y-<\/em>intercept is [latex]\\frac{1}{3}[\/latex], but that really does not enter into our problem, as the only thing we need for two lines to be parallel is the same slope.\r\n\r\nThe one exception is that if the <em>y-<\/em>intercepts are the same, then the two lines are the same line. The next step is to use this slope and the given point in point-slope form.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}y - 5=-\\frac{5}{3}\\left(x - 3\\right)\\hfill \\\\ y - 5=-\\frac{5}{3}x+5\\hfill \\\\ y=-\\frac{5}{3}x+10\\hfill \\end{array}[\/latex]<\/p>\r\nThe equation of the line is [latex]y=-\\frac{5}{3}x+10[\/latex].\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/17232111\/CNX_CAT_Figure_02_02_008.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 2 to 12 in intervals of 2. Two functions are graphed on the same plot: y = negative 5 times x\/3 plus 1\/3 and y = negative 5 times x\/3 plus 10. The lines do not cross.\" width=\"487\" height=\"329\" \/> Graph of y = -5\/3 x + 10 and y = -5\/3x + 1\/3[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Find the value of a variable that satisfies an equation<\/li>\n<li>Write equations for lines using different forms: slope-intercept, point-slope, and standard form<\/li>\n<li>Recognize and write equations for horizontal and vertical lines<\/li>\n<li>Determine if lines are parallel or perpendicular, and write equations for lines parallel or perpendicular to a given line<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Solving Linear Equations in One Variable<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Equation Basics:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Contains an equal sign (=) separating two expressions<\/li>\n<li class=\"whitespace-normal break-words\">May include coefficients, variables, terms, and expressions<\/li>\n<li class=\"whitespace-normal break-words\">Goal: Isolate the variable on one side<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Types of Equations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Simple equations (solved mentally): [latex]2x = 6[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Multi-step equations (require multiple operations): [latex]4(\\frac{1}{3}t + \\frac{1}{2}) = 6[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Equations with variables on both sides: [latex]4x - 6 = 2x + 10[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Properties:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Addition Property of Equality: If [latex]a = b[\/latex], then [latex]a + c = b + c[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Multiplication Property of Equality: If [latex]a = b[\/latex], then [latex]ac = bc[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">These properties maintain the balance of the equation<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solving Strategy:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Simplify expressions on both sides<\/li>\n<li class=\"whitespace-normal break-words\">Isolate variable terms on one side<\/li>\n<li class=\"whitespace-normal break-words\">Isolate the variable itself<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Common Techniques:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Combining like terms<\/li>\n<li class=\"whitespace-normal break-words\">Clearing fractions or decimals<\/li>\n<li class=\"whitespace-normal break-words\">Moving terms between sides of the equation<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Solving Process<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Simplify each side of the equation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Clear parentheses<\/li>\n<li class=\"whitespace-normal break-words\">Combine like terms<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Move variable terms to one side:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Add or subtract to move terms<\/li>\n<li class=\"whitespace-normal break-words\">Choose the side with the larger coefficient (if applicable)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Move constant terms to the other side:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Add or subtract as needed<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Isolate the variable:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Multiply or divide both sides<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Check the solution:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Substitute the result back into the original equation<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Solve the equation: [latex]2(3x - 4) - 5 = 4x + 7[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q610206\">Show Answer<\/button><\/p>\n<div id=\"q610206\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li class=\"whitespace-normal break-words\">Simplify the left side by distributing: [latex]2(3x - 4) - 5 = 4x + 7[\/latex] [latex]6x - 8 - 5 = 4x + 7[\/latex] [latex]6x - 13 = 4x + 7[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Move variable terms to one side (left) using the Addition Property of Equality: [latex]6x - 4x = 7 + 13[\/latex] [latex]2x = 20[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Isolate the variable using the Multiplication Property of Equality: [latex]x = \\frac{20}{2} = 10[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Check the solution: Left side: [latex]2(3(10) - 4) - 5 = 2(30 - 4) - 5 = 2(26) - 5 = 52 - 5 = 47[\/latex] Right side: [latex]4(10) + 7 = 40 + 7 = 47[\/latex] The solution [latex]x = 10[\/latex] satisfies the original equation.<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\"><\/div>\n<\/div>\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-effhggcc-Wj2WCzzEL28\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Wj2WCzzEL28?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-effhggcc-Wj2WCzzEL28\" 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=13881110&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-effhggcc-Wj2WCzzEL28&#38;vembed=0&#38;video_id=Wj2WCzzEL28&#38;video_target=tpm-plugin-effhggcc-Wj2WCzzEL28\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/How+to+solve+Linear+Equations+with+variables+on+both+sides_+Linear+Algebra+Education_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Solve Linear Equations With Variables on Both Sides : Linear Algebra Education\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Linear Equations<\/h2>\n<h3>Slope-Intercept Form<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Equation form: [latex]y = mx + b[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]m[\/latex] represents the slope<\/li>\n<li class=\"whitespace-normal break-words\">[latex]b[\/latex] represents the y-intercept<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Slope ([latex]m[\/latex]):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Measures the steepness of the line<\/li>\n<li class=\"whitespace-normal break-words\">Calculated as [latex]m = \\frac{\\text{rise}}{\\text{run}} = \\frac{y_2 - y_1}{x_2 - x_1}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Positive slope: line rises from left to right<\/li>\n<li class=\"whitespace-normal break-words\">Negative slope: line falls from left to right<\/li>\n<li class=\"whitespace-normal break-words\">Zero slope: horizontal line<\/li>\n<li class=\"whitespace-normal break-words\">Undefined slope: vertical line<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Y-intercept ([latex]b[\/latex]):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Point where the line crosses the y-axis<\/li>\n<li class=\"whitespace-normal break-words\">Occurs when [latex]x = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Represents the starting point of the line on the y-axis<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Interpreting the equation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]m[\/latex] tells how much [latex]y[\/latex] changes for each unit increase in [latex]x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]b[\/latex] tells the value of [latex]y[\/latex] when [latex]x = 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Given [latex]m=4[\/latex], find the equation of the line in slope-intercept form passing through the point [latex]\\left(2,5\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q634647\">Show Solution<\/button><\/p>\n<div id=\"q634647\" class=\"hidden-answer\" style=\"display: none\">[latex]y=4x - 3[\/latex]<\/div>\n<\/div>\n<\/section>\n<h3>The Point-Slope Formula<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Equation form: [latex]y - y_1 = m(x - x_1)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]m[\/latex] is the slope of the line<\/li>\n<li class=\"whitespace-normal break-words\">[latex](x_1, y_1)[\/latex] is a known point on the line<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Components:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Slope ([latex]m[\/latex]): Rate of change of the line<\/li>\n<li class=\"whitespace-normal break-words\">Point [latex](x_1, y_1)[\/latex]: Any known point on the line<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Usage:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Useful when given a point and slope<\/li>\n<li class=\"whitespace-normal break-words\">Can be easily converted to slope-intercept form<\/li>\n<li class=\"whitespace-normal break-words\">Often used in calculus for tangent line equations<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Flexibility:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Any point on the line can be used as [latex](x_1, y_1)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Allows for easy verification of points on the line<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the equation of a line with slope [latex]m = -2[\/latex] passing through the point [latex](3, 5)[\/latex]. Give your answer in point-slope form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q492478\">Show Answer<\/button><\/p>\n<div id=\"q492478\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li class=\"whitespace-normal break-words\">Identify the components:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Slope: [latex]m = -2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Point: [latex](x_1, y_1) = (3, 5)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Substitute into the point-slope formula: [latex]y - y_1 = m(x - x_1)[\/latex] [latex]y - 5 = -2(x - 3)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">This is the equation in point-slope form. To convert to slope-intercept form: [latex]\\begin{array}{rcl} y - 5 &=& -2(x - 3) \\[2ex] y - 5 &=& -2x + 6 \\[2ex] y &=& -2x + 11 \\end{array}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\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-acdcgcgb-SemcMTLjSiw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/SemcMTLjSiw?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-acdcgcgb-SemcMTLjSiw\" 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=12779085&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-acdcgcgb-SemcMTLjSiw&#38;vembed=0&#38;video_id=SemcMTLjSiw&#38;video_target=tpm-plugin-acdcgcgb-SemcMTLjSiw\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Finding+the+Equation+of+a+Line+Given+Slope+and+a+Point+-+Point+-+Slope+Form_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFinding the Equation of a Line Given Slope and a Point &#8211; Point &#8211; Slope Form\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Standard Form of a Line<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Equation form: [latex]Ax + By = C[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]A[\/latex], [latex]B[\/latex], and [latex]C[\/latex] are integers<\/li>\n<li class=\"whitespace-normal break-words\">[latex]A[\/latex] and [latex]B[\/latex] are not both zero<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Components:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]A[\/latex]: Coefficient of x<\/li>\n<li class=\"whitespace-normal break-words\">[latex]B[\/latex]: Coefficient of y<\/li>\n<li class=\"whitespace-normal break-words\">[latex]C[\/latex]: Constant term<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Characteristics:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">All terms are on one side of the equation<\/li>\n<li class=\"whitespace-normal break-words\">x and y terms are on the left, constant on the right<\/li>\n<li class=\"whitespace-normal break-words\">Coefficients are typically reduced to lowest terms<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Conversions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Can be converted from\/to slope-intercept form<\/li>\n<li class=\"whitespace-normal break-words\">Can be converted from\/to point-slope form<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the equation of the line in standard form with slope [latex]m=-\\frac{1}{3}[\/latex] which passes through the point [latex]\\left(1,\\frac{1}{3}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q3712\">Show Solution<\/button><\/p>\n<div id=\"q3712\" class=\"hidden-answer\" style=\"display: none\">[latex]x+3y=2[\/latex]<\/div>\n<\/div>\n<\/section>\n<section aria-label=\"Example\"><\/section>\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-hgadahda-q88S98Y_Pp0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/q88S98Y_Pp0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hgadahda-q88S98Y_Pp0\" 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=12779135&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-hgadahda-q88S98Y_Pp0&#38;vembed=0&#38;video_id=q88S98Y_Pp0&#38;video_target=tpm-plugin-hgadahda-q88S98Y_Pp0\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Write+Standard+Form+(when+given+point+and+slope)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWrite Standard Form (when given point and slope)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/section>\n<section class=\"textbox example\">Find the equation of the line that passes through [latex](1,2)[\/latex] and [latex](3,10)[\/latex] in slope-intercept, point-slope, and standard form.[latex]\\begin{align*} \\text{Given Points: } & (1, 2) \\text{ and } (3, 10)\\\\ \\text{Calculate Slope: } & \\\\ & m = \\frac{10 - 2}{3 - 1} = \\frac{8}{2} = 4 \\\\ \\text{Point-Slope Form: } & \\\\ & y - 2 = 4(x - 1) \\\\ \\text{Expand to Slope-Intercept Form: } & \\\\ & y - 2 = 4x - 4 \\\\ & y = 4x - 2 \\\\ \\text{Convert to Standard Form: } & \\\\ & -4x + y = -2 \\\\ & 4x - y = 2 \\quad \\text{(Make \\(A\\) positive)} \\end{align*}[\/latex]<\/section>\n<h2>Vertical and Horizontal Lines<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Vertical Lines:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Equation: [latex]x = c[\/latex], where [latex]c[\/latex] is a constant<\/li>\n<li class=\"whitespace-normal break-words\">Slope: Undefined<\/li>\n<li class=\"whitespace-normal break-words\">All points have the same x-coordinate<\/li>\n<li class=\"whitespace-normal break-words\">Cannot be expressed in slope-intercept form<\/li>\n<li class=\"whitespace-normal break-words\">Represent a constant [latex]x[\/latex]-value for all [latex]y[\/latex]-values<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal Lines:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Equation: [latex]y = c[\/latex], where [latex]c[\/latex] is a constant<\/li>\n<li class=\"whitespace-normal break-words\">Slope: Zero<\/li>\n<li class=\"whitespace-normal break-words\">All points have the same [latex]y[\/latex]-coordinate<\/li>\n<li class=\"whitespace-normal break-words\">Can be expressed in slope-intercept form with [latex]m = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Represent a constant [latex]y[\/latex]-value for all [latex]x[\/latex]-values<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Characteristics:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Vertical lines are parallel to the y-axis<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal lines are parallel to the x-axis<\/li>\n<li class=\"whitespace-normal break-words\">These lines are perpendicular to each other<\/li>\n<\/ul>\n<\/li>\n<li>Identifying Line Type:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li>If all points share the same [latex]x[\/latex]-coordinate: Vertical line<\/li>\n<li>If all points share the same [latex]y[\/latex]-coordinate: Horizontal line<\/li>\n<li>If neither: Oblique (slanted) line<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the equation of the line passing through the given points: [latex]\\left(1,-3\\right)[\/latex] and [latex]\\left(1,4\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q122244\">Show Solution<\/button><\/p>\n<div id=\"q122244\" class=\"hidden-answer\" style=\"display: none\">The [latex]x[\/latex]<em>&#8211;<\/em>coordinate of both points is [latex]1[\/latex]. Therefore, we have a vertical line, [latex]x=1[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the equation of the line passing through [latex]\\left(-5,2\\right)[\/latex] and [latex]\\left(2,2\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q864962\">Show Solution<\/button><\/p>\n<div id=\"q864962\" class=\"hidden-answer\" style=\"display: none\">Horizontal line: [latex]y=2[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Parallel and Perpendicular Lines<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li>Parallel Lines:\n<ul>\n<li>Equation: Lines have the same slope but different y-intercepts.<\/li>\n<li>Slope: Same for all parallel lines.<\/li>\n<li>Properties:\n<ul>\n<li>Parallel lines never intersect.<\/li>\n<li>The distance between the lines is constant (equidistant).<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>Perpendicular Lines:\n<ul>\n<li>Equation: Lines have slopes that are negative reciprocals of each other.<\/li>\n<li>Slope: The product of the slopes of two perpendicular lines is -1.<\/li>\n<li>Properties:\n<ul>\n<li>Perpendicular lines intersect at a 90-degree angle.<\/li>\n<li>If one line has slope [latex]m[\/latex], the perpendicular line will have slope [latex]-\\dfrac{1}{m}[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>Identifying Line Type:\n<ul>\n<li>Parallel Lines: Look for lines with identical slopes and different y-intercepts.<\/li>\n<li>Perpendicular Lines: Check if the product of their slopes is -1.<\/li>\n<\/ul>\n<\/li>\n<li>Writing equations of parallel and perpendicular lines\n<ul>\n<li>To write the equation of a line <strong>parallel<\/strong> to a given line, use the same slope as the original line but adjust the y-intercept based on a point the line passes through.<\/li>\n<li>To write the equation of a line <strong>perpendicular<\/strong> to a given line, use the negative reciprocal of the original slope and find the y-intercept using a point the line passes through.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">Graph the two lines and determine whether they are parallel, perpendicular, or neither: [latex]2y-x=10[\/latex] and [latex]2y=x+4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q727314\">Show Solution<\/button><\/p>\n<div id=\"q727314\" class=\"hidden-answer\" style=\"display: none\">Parallel lines. Write the equations in slope-intercept form.<\/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-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200328\/CNX_CAT_Figure_02_02_007.jpg\" alt=\"Coordinate plane with the x-axis ranging from negative 5 to 5 and the y-axis ranging from negative 1 to 6. Two functions are graphed on the same plot: y = x\/2 plus 5 and y = x\/2 plus 2. The lines do not cross.\" width=\"487\" height=\"329\" \/><figcaption class=\"wp-caption-text\">Graph of y = 1\/2x + 5 and y = 1\/2x + 2<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">A line passes through the points [latex](\u20132, 6)[\/latex] and [latex](4, 5)[\/latex]. Find the equation of a perpendicular line that passes through the point [latex](4, 5)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q119051\">Show Solution<\/button><\/p>\n<div id=\"q119051\" class=\"hidden-answer\" style=\"display: none\">\nFrom the two points of the given line, we can calculate the slope of that line.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}{m}_{1}=\\frac{5 - 6}{4-\\left(-2\\right)}\\hfill & =\\frac{-1}{6}\\hfill & =-\\frac{1}{6}\\hfill \\end{array}[\/latex]<\/p>\n<p>Find the negative reciprocal of the slope.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}{m}_{2}=\\frac{-1}{-\\frac{1}{6}}\\hfill & =-1\\left(-\\frac{6}{1}\\right)\\hfill & =6\\hfill \\end{array}[\/latex]<\/p>\n<p>We can then solve for the <em>y-<\/em>intercept of the line passing through the point (4, 5).<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lllll}y=6x+b\\hfill \\\\ 5=6\\left(4\\right)+b\\hfill \\\\ 5=24+b\\hfill \\\\ -19=b\\hfill \\\\ b=-19\\hfill \\end{array}[\/latex]<\/p>\n<p>The equation for the line that is perpendicular to the line passing through the two given points and also passes through point (4, 5) is:<\/p>\n<p style=\"text-align: center;\">[latex]y=6x - 19[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Write the equation of line parallel to a [latex]5x+3y=1[\/latex] which passes through the point [latex]\\left(3,5\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q72460\">Show Solution<\/button><\/p>\n<div id=\"q72460\" class=\"hidden-answer\" style=\"display: none\">First, we will write the equation in slope-intercept form to find the slope.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}5x+3y=1\\hfill \\\\ 3y=-5x+1\\hfill \\\\ y=-\\frac{5}{3}x+\\frac{1}{3}\\hfill \\end{array}[\/latex]<\/p>\n<p>The slope is [latex]m=-\\frac{5}{3}[\/latex]. The <em>y-<\/em>intercept is [latex]\\frac{1}{3}[\/latex], but that really does not enter into our problem, as the only thing we need for two lines to be parallel is the same slope.<\/p>\n<p>The one exception is that if the <em>y-<\/em>intercepts are the same, then the two lines are the same line. The next step is to use this slope and the given point in point-slope form.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}y - 5=-\\frac{5}{3}\\left(x - 3\\right)\\hfill \\\\ y - 5=-\\frac{5}{3}x+5\\hfill \\\\ y=-\\frac{5}{3}x+10\\hfill \\end{array}[\/latex]<\/p>\n<p>The equation of the line is [latex]y=-\\frac{5}{3}x+10[\/latex].<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/17232111\/CNX_CAT_Figure_02_02_008.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 2 to 12 in intervals of 2. Two functions are graphed on the same plot: y = negative 5 times x\/3 plus 1\/3 and y = negative 5 times x\/3 plus 10. The lines do not cross.\" width=\"487\" height=\"329\" \/><figcaption class=\"wp-caption-text\">Graph of y = -5\/3 x + 10 and y = -5\/3x + 1\/3<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":12,"menu_order":19,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Finding the Equation of a Line Given Slope and a Point - Point - Slope Form \",\"author\":\"\",\"organization\":\"MATH TEACHER GON\",\"url\":\"https:\/\/www.youtube.com\/watch?v=SemcMTLjSiw\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Write Standard Form (when given point and slope) \",\"author\":\"\",\"organization\":\"AlgebraConceptVideos\",\"url\":\"https:\/\/www.youtube.com\/watch?v=q88S98Y_Pp0\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"How to solve Linear Equations with variables on both sides: Linear Algebra Education \",\"author\":\"\",\"organization\":\"LKLogic\",\"url\":\"https:\/\/www.youtube.com\/watch?v=wShnYemIr28\",\"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":75,"module-header":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"Finding the Equation of a Line Given Slope and a Point - 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