{"id":28866,"date":"2021-04-20T13:00:28","date_gmt":"2021-04-20T13:00:28","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28866"},"modified":"2024-09-22T02:07:32","modified_gmt":"2024-09-22T02:07:32","slug":"equation-of-the-line-and-cartesian-systems","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/equation-of-the-line-and-cartesian-systems\/","title":{"rendered":"Equation of the Line and Cartesian Systems"},"content":{"rendered":"<p><center><\/p>\n<h1>Equation of the Line and Cartesian Systems<\/h1>\n<p style=\"text-align: center;\"><em><strong>Summary:<\/strong><br \/>\n      In this class, we will cover the fundamentals of analytic geometry, showing how to represent points on a plane using coordinates and how to formulate the equation of the line from the slope and a given point. Key concepts such as the slope, the use of the equation <span class=\"katex-eq\" data-katex-display=\"false\">y = mx + b<\/span>, and the graphical representation of lines are explored, along with practical exercises and applications to solve real-world problems, such as calculating positions and line intersections.<\/em><\/p>\n<p>   <strong>Learning Objectives<\/strong><\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Understand<\/strong> the basic principles of analytic geometry and its application in representing points on a Cartesian plane.<\/li>\n<li><strong>Identify<\/strong> the formula for the slope of a line and its geometric meaning.<\/li>\n<li><strong>Apply<\/strong> the general equation of the line <span class=\"katex-eq\" data-katex-display=\"false\">y = mx + b<\/span> to describe linear relationships.<\/li>\n<li><strong>Calculate<\/strong> the equation of the line from a point and the slope.<\/li>\n<li><strong>Graph<\/strong> lines on a Cartesian plane using their linear equation.<\/li>\n<li><strong>Solve<\/strong> problems involving the intersection of two lines using systems of equations.<\/li>\n<li><strong>Analyze<\/strong> the relationship between two linear magnitudes and how to represent them through a line equation.<\/li>\n<\/ol>\n<p>   <strong>CONTENT INDEX<\/strong><br \/>\n   <a href=\"#1\">The Principles of Analytic Geometry<\/a><br \/>\n   <a href=\"#2\">The Equation of the Line<\/a><br \/>\n   <a href=\"#3\">How to Graph the Equation of the Line<\/a><br \/>\n   <a href=\"#4\">Line Intersections<\/a>\n   <\/p>\n<p><\/center><\/p>\n<p>   <center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/mNISGHOByAI\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p style=\"text-align: justify;\">We will now begin our study of the equation of the line, Cartesian systems, and the principles of analytic geometry.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>The Principles of Analytic Geometry<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=140s\" target=\"_blank\" rel=\"noopener\"><strong>When real numbers are introduced<\/strong><\/a>, it is usually said that these are points on a line<\/p>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-GVaIW2wJ8sQ\/YH8dbLc562I\/AAAAAAAAE8Q\/mQOvNS6N18gbsEvWuU4gzrjvBYuS20_-ACLcBGAsYHQ\/s0\/RECTA%2BDE%2BLOS%2BREALES.PNG\" alt=\"REAL LINE\" class=\" aligncenter lazyload\" width=\"581\" height=\"128\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-GVaIW2wJ8sQ\/YH8dbLc562I\/AAAAAAAAE8Q\/mQOvNS6N18gbsEvWuU4gzrjvBYuS20_-ACLcBGAsYHQ\/s0\/RECTA%2BDE%2BLOS%2BREALES.PNG\" alt=\"REAL LINE\" class=\" aligncenter lazyload\" width=\"581\" height=\"128\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Based on this, Descartes had the genius idea of using two lines to represent points on a plane as a pair of coordinates (x, y)<\/p>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-HVR3PImxths\/YH8d4-azX3I\/AAAAAAAAE8Y\/Qw5_Ir_CzZwM446x73emMv2R4FRssqfHwCLcBGAsYHQ\/s0\/PLANO%2BCARTESIANO.PNG\" alt=\"CARTESIAN PLANE\" class=\" aligncenter lazyload\" width=\"434\" height=\"252\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-HVR3PImxths\/YH8d4-azX3I\/AAAAAAAAE8Y\/Qw5_Ir_CzZwM446x73emMv2R4FRssqfHwCLcBGAsYHQ\/s0\/PLANO%2BCARTESIANO.PNG\" alt=\"CARTESIAN PLANE\" class=\" aligncenter lazyload\" width=\"434\" height=\"252\" \/><\/noscript><\/p>\n<p><a name=\"2\"><\/a>     <\/p>\n<h2>The Equation of the Line<\/h2>\n<p style=\"text-align: justify;\">Using these concepts, it is now possible to consider a set of points on the plane to form curves, where the coordinate <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> corresponds to another coordinate <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span>, and this correspondence rule is given by a function. This is where algebra penetrates geometry, giving birth to \u00abAnalytic Geometry.\u00bb<\/p>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=320s\" target=\"_blank\" rel=\"noopener\"><strong>Geometrically, we understand a line<\/strong><\/a> as the curve that connects two points by traveling the shortest possible distance.<\/p>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-Bgjb959iAoo\/YH8eKEZ_ocI\/AAAAAAAAE8g\/FaZmlsj4Pn8tZ_A_XpqA5yfE7SWdygj7QCLcBGAsYHQ\/s0\/RECTA%2BEN%2BEL%2BPLANO%2BCARTESIANO.PNG\" alt=\"LINE IN THE CARTESIAN PLANE\" class=\" aligncenter lazyload\" width=\"429\" height=\"267\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-Bgjb959iAoo\/YH8eKEZ_ocI\/AAAAAAAAE8g\/FaZmlsj4Pn8tZ_A_XpqA5yfE7SWdygj7QCLcBGAsYHQ\/s0\/RECTA%2BEN%2BEL%2BPLANO%2BCARTESIANO.PNG\" alt=\"LINE IN THE CARTESIAN PLANE\" class=\" aligncenter lazyload\" width=\"429\" height=\"267\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Geometrically, we understand a line as the curve that connects two points by traveling the shortest possible distance. Analyzing this, by virtue of Thales&#8217; theorem, we will see that for every increment of the coordinate <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span>, there is a corresponding increment of the coordinate <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> such that the quotient <span class=\"katex-eq\" data-katex-display=\"false\">m=(y_2 - y_1)\/(x_2 - x_1)=\\Delta y \/ \\Delta x<\/span> will always be constant for any pair of points on the line. This is what we call the <strong>\u00abslope of the line.\u00bb<\/strong><\/p>\n<p style=\"text-align: justify;\">Since the slope is the same for any pair of points on the line, if we consider points on the line with coordinates <span class=\"katex-eq\" data-katex-display=\"false\">(x, y),<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">(x_0, y_0),<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">(x_1, y_1)<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">(x_2, y_2),<\/span> we can write:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{y - y_0}{x - x_0} = \\frac{y_2 - y_1}{x_2 - x_1}<\/span>\n<p style=\"text-align: justify;\">Which is the same as saying<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{matrix}y &amp; = &amp; \\displaystyle \\frac{y_2 - y_1}{x_2 - x_1} (x - x_0 ) + y_0 \\\\ \\\\ &amp; = &amp; \\displaystyle \\frac{\\Delta y}{\\Delta x} (x - x_0) + y_0 \\end{matrix}<\/span>\n<p style=\"text-align: justify;\">This is where the well-known <strong>equation of the line<\/strong> comes from<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\color{red}{{y = m(x - x_0) + y_0}}<\/span>\n<p style=\"text-align: justify;\">Here, the pair <span class=\"katex-eq\" data-katex-display=\"false\">(x_0, y_0)<\/span> is a fixed point, while the pair <span class=\"katex-eq\" data-katex-display=\"false\">(x, y)<\/span> is any point.<\/p>\n<h3>Example Exercises<\/h3>\n<ol style=\"text-align: justify;\">\n<li>Calculate the equation of the line that passes through the point <span class=\"katex-eq\" data-katex-display=\"false\">(x_0, y_0) = (2, 3)<\/span> with a slope of <span class=\"katex-eq\" data-katex-display=\"false\">m = 3\/2<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=695s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTION]<\/strong><\/a><\/li>\n<li>Calculate the equation of the line that passes through the point <span class=\"katex-eq\" data-katex-display=\"false\">(x_0, y_0) = (1, 8)<\/span> with a slope of <span class=\"katex-eq\" data-katex-display=\"false\">m = 7\/5<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=750s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTION]<\/strong><\/a><\/li>\n<li>Calculate the equation of the line that passes through the points <span class=\"katex-eq\" data-katex-display=\"false\">(x_1, y_1) = (3, 5)<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">(x_2, y_2) = (1, -2)<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=818s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTION]<\/strong><\/a><\/li>\n<\/ol>\n<p><a name=\"3\"><\/a><\/p>\n<h2>How to Graph the Equation of the Line<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=1063s\" target=\"_blank\" rel=\"noopener\"><strong>We have already seen how to obtain<\/strong><\/a> the equation of the line from some graphical information; now we will follow the reverse path, obtaining the graphical representation from the equation of the line.<\/p>\n<p style=\"text-align: justify;\">At the end of the day, the equation of the line always ends up being presented as follows:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">y = mx + b<\/span>\n<p style=\"text-align: justify;\">Where <span class=\"katex-eq\" data-katex-display=\"false\">m = \\Delta Y \/ \\Delta x<\/span> is the slope and <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> is the position coefficient. From this, we have the following figure<\/p>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-qJd27JlFvC8\/YH8eiEun69I\/AAAAAAAAE8o\/_swD6Cx2J_gQLY5Nw8RoSo7cPfmaOpzLgCLcBGAsYHQ\/s0\/RECTA%2BEN%2BEL%2BPLANO%2BCARTESIANO%2BCON%2BCOORDENADAS.PNG\" alt=\"LINE IN THE CARTESIAN PLANE WITH COORDINATES\" class=\" aligncenter lazyload\" width=\"350\" height=\"264\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-qJd27JlFvC8\/YH8eiEun69I\/AAAAAAAAE8o\/_swD6Cx2J_gQLY5Nw8RoSo7cPfmaOpzLgCLcBGAsYHQ\/s0\/RECTA%2BEN%2BEL%2BPLANO%2BCARTESIANO%2BCON%2BCOORDENADAS.PNG\" alt=\"LINE IN THE CARTESIAN PLANE WITH COORDINATES\" class=\" aligncenter lazyload\" width=\"350\" height=\"264\" \/><\/noscript><\/p>\n<h3>Example Exercise<\/h3>\n<ol style=\"text-align: justify;\">\n<li>Graph the line with the equation <span class=\"katex-eq\" data-katex-display=\"false\">y = \\displaystyle \\frac{3}{4}x + 2<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=1139s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTION]<\/strong><\/a><\/li>\n<li>Graph the line with the equation <span class=\"katex-eq\" data-katex-display=\"false\">y = \\displaystyle -\\frac{2}{5}x + 6<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=1209s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTION]<\/strong><\/a><\/li>\n<\/ol>\n<h3>Application Problems for the Line Equation<\/h3>\n<p style=\"text-align: justify;\">The line can be used to solve problems that involve the direct relationship between two magnitudes, as in the following examples:<\/p>\n<ol style=\"text-align: justify;\">\n<li>A vehicle with an initial position <span class=\"katex-eq\" data-katex-display=\"false\">x_0 = 12[m]<\/span> moves with a speed of <span class=\"katex-eq\" data-katex-display=\"false\">v = 0.3[m\/s]<\/span>. What will its position be after <span class=\"katex-eq\" data-katex-display=\"false\">30[s]<\/span>? <a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=1257s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTION]<\/strong><\/a><\/li>\n<li>A person goes to the market and buys <span class=\"katex-eq\" data-katex-display=\"false\">1[kg]<\/span> of apples, spending a total of <span class=\"katex-eq\" data-katex-display=\"false\">50 Z\\$.<\/span> That same day, the same person returned to the market to buy another <span class=\"katex-eq\" data-katex-display=\"false\">3[kg]<\/span> of apples, spending a total of <span class=\"katex-eq\" data-katex-display=\"false\">60 Z\\$.<\/span>. What is the price of the apples, and what is the price of the tickets?<a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=1383s\" target=\"_blank\" rel=\"noopener\"><strong> [SOLUTION]<\/strong><\/a><\/li>\n<\/ol>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Line Intersections<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=1855s\" target=\"_blank\" rel=\"noopener\"><strong>Suppose we have two lines<\/strong><\/a> and we want to find the common point between them; that is, find the intersection of the lines. To solve these types of problems, we must solve a system of equations. To understand this better, let&#8217;s look at the following example.<\/p>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=1914s\" target=\"_blank\" rel=\"noopener\"><strong>Let&#8217;s consider the following lines:<\/strong><\/a><\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">L_1 \\; : \\; y= \\displaystyle \\frac{3}{2}x + 1<\/span>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">L_1 \\; : \\; y=\\displaystyle -\\frac{1}{3}x + 9<\/span>\n<p style=\"text-align: justify;\">Where do these two lines intersect?<\/p>\n<p style=\"text-align: justify;\">To solve this, we reason as follows:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\displaystyle \\frac{3}{2}x + 1<\/span><\/td>\n<td>; Line <span class=\"katex-eq\" data-katex-display=\"false\">L_1<\/span><\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y= \\displaystyle -\\frac{1}{3}x + 9<\/span><\/td>\n<td>; Line <span class=\"katex-eq\" data-katex-display=\"false\">L_2<\/span><\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{3}{2}x + 1 = -\\frac{1}{3}x + 9<\/span><\/td>\n<td>; From (1) and (2)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{3}{2}x = -\\frac{1}{3}x + 8<\/span><\/td>\n<td>; Subtracting 1 from both sides<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">9x = -2x + 48<\/span><\/td>\n<td>; Multiplying both sides by 6<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">11x =48<\/span><\/td>\n<td>; Adding 2x to both sides<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle x = \\frac{48}{11}<\/span><\/td>\n<td>; Dividing both sides by 11<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle y= \\frac{3}{2}\\cdot \\frac{48}{11} + 1<\/span><\/td>\n<td>; From (1) and (3)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle y= \\frac{3}{1}\\cdot \\frac{24}{11} + \\frac{11}{11}<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y= \\displaystyle \\frac{83}{11}<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle (x, y)= \\left(\\frac{48}{11}, \\frac{83}{11} \\right)<\/span><\/td>\n<td>; From (3) and (4)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Therefore, the intersection point between the lines is <span class=\"katex-eq\" data-katex-display=\"false\">(x, y)= \\displaystyle \\left(\\frac{48}{11}, \\frac{83}{11} \\right).<\/span>\n<h3>Example Application Problem for Line Intersections<\/h3>\n<p style=\"text-align: justify;\">For a party, a total of 600 tickets were sold with total revenue of <span class=\"katex-eq\" data-katex-display=\"false\">\\$1,300,000.<\/span> Youth tickets were sold for <span class=\"katex-eq\" data-katex-display=\"false\">\\$1,000,<\/span> and adult tickets for <span class=\"katex-eq\" data-katex-display=\"false\">\\$3,000.<\/span> How many adults and youth attended the party?<a href=\"https:\/\/www.youtube.com\/watch?v=mNISGHOByAI&amp;t=2255s\" target=\"_blank\" rel=\"noopener\"><strong> [SOLUTION]<\/strong><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Equation of the Line and Cartesian Systems Summary: In this class, we will cover the fundamentals of analytic geometry, showing how to represent points on a plane using coordinates and how to formulate the equation of the line from the slope and a given point. Key concepts such as the slope, the use of the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28865,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":6,"footnotes":""},"categories":[583,567],"tags":[],"class_list":["post-28866","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-and-geometry","category-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Equation of the Line and Cartesian Systems - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Learn the fundamentals of the Equation of the Line and Cartesian Systems. 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