{"id":2686,"date":"2016-04-15T05:09:03","date_gmt":"2016-04-15T05:09:03","guid":{"rendered":"https:\/\/courses.candelalearning.com\/nrocarithmetic\/?post_type=chapter&#038;p=2686"},"modified":"2018-01-03T23:54:14","modified_gmt":"2018-01-03T23:54:14","slug":"read-or-watch-slope-intercept-form-of-a-line","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-beginalgebra\/chapter\/read-or-watch-slope-intercept-form-of-a-line\/","title":{"raw":"Writing Equations of Lines","rendered":"Writing Equations of Lines"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Write the equation and draw the graph of a line using slope and y-intercept\r\n<ul>\r\n \t<li>Write the equation of a line using slope and y-intercept<\/li>\r\n \t<li>Rearrange a linear equation so it is in slope-intercept form.<\/li>\r\n \t<li>Graph a line using slope and y-intercept<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Write and solve equations of lines using slope and a point on the line\r\n<ul>\r\n \t<li>Write the equation of a line given the slope and a point on the line.<\/li>\r\n \t<li>Identify which parts of a linear equation are given and which parts need to be solved for using algebra<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Write and solve equations of lines using two points on the line\r\n<ul>\r\n \t<li>Write the equation of a line given two points on the line<\/li>\r\n \t<li>Identify which parts of a linear equation are given and which parts need to be solved for using algebra.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Write equations of parallel and perpendicular lines\r\n<ul>\r\n \t<li>Find a line that is parallel to another line given a point<\/li>\r\n \t<li>Find a line that is perpendicular to another line given a point<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Interpret the y-intercept of a linear equation and use that equation to make predictions\r\n<ul>\r\n \t<li>Interpret the <em>y<\/em>-intercept of a linear equation<\/li>\r\n \t<li>Use a linear equation to make a prediction<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\nWhen graphing a line we found one method we could use is to make a table of values. However, if we can identify some properties of the line, we may be able to make a graph much quicker and easier. One such method is finding the slope and the <em>y<\/em>-intercept of the equation. The slope can be represented by m and the <em>y<\/em>-intercept, where it crosses the axis and [latex]x=0[\/latex], can be represented by [latex](0,b)[\/latex] where <em>b<\/em> is the value where the graph crosses the vertical <em>y<\/em>-axis. Any other point on the line can be represented by [latex](x,y)[\/latex].\r\n\r\nIn the equation,\r\n<p style=\"text-align: center;\">[latex]y = mx + b[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,m\\,\\,\\,\\,=\\,\\,\\,\\text{slope}\\\\(x,y)=\\,\\,\\,\\text{a point on the line}\\\\\\,\\,\\,\\,\\,\\,\\,b\\,\\,\\,\\,=\\,\\,\\,\\text{the y value of the y-intercept}\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">This formula is known as the slope-intercept equation.\u00a0If we know the slope and the <em>y<\/em>-intercept we can easily find the equation that represents the line<\/p>\r\n\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWrite the equation of the line that has a slope of [latex] \\displaystyle \\frac{1}{2}[\/latex] and a <i>y<\/i>-intercept of [latex]\u22125[\/latex].\r\n\r\n[reveal-answer q=\"624715\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"624715\"]Substitute the slope (<i>m<\/i>) into [latex]y=mx+b[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\displaystyle y=\\frac{1}{2}x+b[\/latex]<\/p>\r\nSubstitute the <i>y<\/i>-intercept (<i>b<\/i>) into the equation.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle y=\\frac{1}{2}x-5[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]y=\\frac{1}{2}x-5[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe can also find the equation by looking at a graph and finding the slope and <em>y<\/em>-intercept.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWrite the equation of the line in the graph by identifying the slope and <em>y<\/em>-intercept.\r\n<img class=\"size-medium wp-image-3198 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/26035543\/SVG_Grapher-300x297.png\" alt=\"SVG_Grapher\" width=\"300\" height=\"297\" \/>\r\n[reveal-answer q=\"96446\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"96446\"]Identify the point where the graph crosses the y-axis [latex](0,3)[\/latex]. This means the <em>y<\/em>-intercept is 3.\r\n\r\nIdentify one other point and draw a slope triangle to find the slope.\r\n\r\nThe slope is [latex]\\frac{-2}{3}[\/latex]\r\n\r\nSubstitute the slope and <em>y<\/em> value of the intercept into the slope-intercept equation.\r\n<p style=\"text-align: center;\">[latex]y=mx+b\\\\y=\\frac{-2}{3}x+b\\\\y=\\frac{-2}{3}x+3[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]y=\\frac{-2}{3}x+3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe can also move the opposite direction, using the equation identify the slope\u00a0and <em>y<\/em>-intercept and graph the equation from this information. However, it will be\u00a0important for the equation to first be in slope intercept form. If it is not, we will\u00a0have to solve it for <em>y<\/em> so we can identify the slope and the <em>y<\/em>-intercept.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWrite the following equation in slope-intercept form.\r\n<p style=\"text-align: center;\">[latex]2x+4y=6[\/latex]<\/p>\r\n[reveal-answer q=\"373034\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"373034\"]We need to solve for <em>y<\/em>. Start by subtracting [latex]2[\/latex] from both sides.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2x\\,\\,\\,+\\,\\,\\,4y\\,\\,\\,=\\,\\,\\,6\\\\-2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-2x\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">It helps to place the <em>x<\/em> term first on the right hand side. Notice how we keep the 6 positive by placing an addition sign in front.<\/p>\r\n<p style=\"text-align: center;\">[latex]4y=-2x+6[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Divide each term by 4 to isolate the <em>y<\/em>.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{4y}{4}=\\frac{-2x}{4}+\\frac{6}{4}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]y=\\frac{-2x}{4}+\\frac{6}{4}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Reduce the fractions<\/p>\r\n<p style=\"text-align: center;\">[latex]y=-\\frac{1}{2}x+\\frac{3}{2}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]y=-\\frac{1}{2}x+\\frac{3}{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nOnce we have an equation in slope-intercept form we can graph it by first plotting\u00a0the <em>y<\/em>-intercept, then using the slope, find a second point and connecting the dots.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nGraph [latex]y=\\frac{1}{2}x-4[\/latex] using the slope-intercept equation.\r\n\r\n[reveal-answer q=\"420487\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"420487\"]First, plot the <em>y<\/em>-intercept.\r\n\r\n<img class=\"aligncenter wp-image-3202 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/26042734\/SVG_Grapher2-300x294.png\" alt=\"The y-intercept plotted at negative 4 on the y axis.\" width=\"300\" height=\"294\" \/>\r\n\r\nNow use the slope to count up or down and over left or right to the next point. This slope is [latex]\\frac{1}{2}[\/latex], so you can count up one and right two\u2014both positive because both parts of the slope are positive.\r\n\r\nConnect the dots.\r\n<img class=\"aligncenter wp-image-3203 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/26043304\/SVG_Grapher3-300x289.png\" alt=\"A line crosses through negative 4 on the y-axis and has a slope of 1\/2.\" width=\"300\" height=\"289\" \/>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n<h2>Slope-Intercept Form of a Line<\/h2>\r\nhttps:\/\/youtu.be\/GIn7vbB5AYo\r\n\r\n&nbsp;\r\n<h2 id=\"Find the Equation of a Line Given the Slope and a Point on the Line\">Find the Equation of a Line Given the Slope and a Point on the Line<\/h2>\r\nUsing the slope-intercept equation of a line is possible when you know both the slope (<i>m<\/i>) and the <i>y<\/i>-intercept (<i>b<\/i>), but what if you know the slope and just any point on the line, not specifically the <i>y<\/i>-intercept? Can you still write the equation? The answer is <i>yes<\/i>, but you will need to put in a little more thought and work than you did previously.\r\n\r\nRecall that a point is an (<i>x<\/i>, <i>y<\/i>) coordinate pair and that all points on the line will satisfy the linear equation. So, if you have a point on the line, it must be a solution to the equation. Although you don\u2019t know the exact equation yet, you know that you can express the line in slope-intercept form, [latex]y=mx+b[\/latex].\r\n\r\nYou do know the slope (<i>m<\/i>), but you just don\u2019t know the value of the <i>y<\/i>-intercept (<i>b<\/i>). Since point (<i>x<\/i>, <i>y<\/i>) is a solution to the equation, you can substitute its coordinates for <i>x<\/i> and <i>y<\/i> in [latex]y=mx+b[\/latex]\u00a0and solve to find <i>b<\/i>!\r\n\r\nThis may seem a bit confusing with all the variables, but an example with an actual slope and a point will help to clarify.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWrite the equation of the line that has a slope of 3 and contains the point [latex](1,4)[\/latex].\r\n\r\n[reveal-answer q=\"161353\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"161353\"]\r\n\r\nSubstitute the slope (<i>m<\/i>) into\u00a0[latex]y=mx+b[\/latex].\r\n<p style=\"text-align: center;\">[latex]y=3x+b[\/latex]<\/p>\r\nSubstitute the point [latex](1,4)[\/latex] for <i>x <\/i>and <i>y.<\/i>\r\n<p style=\"text-align: center;\">[latex]4=3\\left(1\\right)+b[\/latex]<\/p>\r\nSolve for <i>b.<\/i>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}4=3+b\\\\1=b\\end{array}[\/latex]<\/p>\r\nRewrite [latex]y=mx+b[\/latex]\u00a0with [latex]m=3[\/latex]\u00a0and [latex]b=1[\/latex].\r\n<h4>Answer<\/h4>\r\n[latex]y=3x+1[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nTo confirm our algebra, you can check by graphing the equation [latex]y=3x+1[\/latex]. The equation checks because when graphed it passes through the point [latex](1,4)[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064327\/image045.jpg\" alt=\"An uphill line passes through the y-intercept of (0,1) and the point (1,4). The rise is 3 and the run is 1.\" width=\"348\" height=\"349\" \/>\r\n\r\nIf you know the slope of a line and a point on the line, you can draw a graph. Using an equation in the point-slope form allows you to identify the slope and a point. Consider the equation [latex] \\displaystyle y=-3x-1[\/latex]. The <em>y<\/em>-intercept is the point on the line where it passes through the <em>y<\/em>-axis. What is the value of <em>x<\/em> at this point?\r\n<div class=\"textbox shaded\">Reminder: All <em>y<\/em>-intercepts are points in the form [latex](0,y)[\/latex]. \u00a0The <em>x<\/em> value of any <em>y<\/em>-intercept is <em>always<\/em>\u00a0zero.<\/div>\r\nTherefore, you can tell from this equation that the <i>y<\/i>-intercept is at [latex](0,\u22121)[\/latex], check this by replacing <em>x<\/em> with 0 and solving for <em>y<\/em>. To graph the line, start by plotting that point, [latex](0,\u22121)[\/latex], on a graph.\r\n\r\nYou can also tell from the equation that the slope of this line is [latex]\u22123[\/latex]. So start at [latex](0,\u22121)[\/latex] and count up 3 and over [latex]\u22121[\/latex] (1 unit in the negative direction, left) and plot a second point. (You could also have gone down 3 and over 1.) Then draw a line through both points, and there it is, the graph of [latex] \\displaystyle y=-3x-1[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064324\/image044.jpg\" alt=\"A downhill line passes through the point (-1,2) and the y-intercept (0,-1). The rise is 3 and the run is -1.\" width=\"325\" height=\"326\" \/>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example (Advanced)<\/h3>\r\nWrite the equation of the line that has a slope of [latex]-\\frac{7}{8}[\/latex]\u00a0and contains the point [latex]\\left(4,\\frac{5}{4}\\right)[\/latex].\r\n\r\n[reveal-answer q=\"31452\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"31452\"]\r\n\r\nSubstitute the slope (<i>m<\/i>) into [latex]y=mx+b[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}y=mx+b\\\\\\\\y=-\\frac{7}{8}x+b\\end{array}[\/latex]<\/p>\r\nSubstitute the point [latex]\\left(4,\\frac{5}{4}\\right)[\/latex]\u00a0for <i>x <\/i>and <i>y.<\/i>\r\n<p style=\"text-align: center;\">[latex]\\frac{5}{4}=-\\frac{7}{8}\\left(4\\right)+b[\/latex]<\/p>\r\nSolve for <i>b.<\/i>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{5}{4}=-\\frac{28}{8}+b\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{5}{4}=-\\frac{14}{4}+b\\\\\\\\\\frac{5}{4}+\\frac{14}{4}=-\\frac{14}{4}+\\frac{14}{4}+b\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{19}{4}=b\\end{array}[\/latex]<\/p>\r\nRewrite [latex]y=mx+b[\/latex] with [latex] \\displaystyle m=-\\frac{7}{8}[\/latex] and [latex] \\displaystyle b=\\frac{19}{4}[\/latex].\r\n<h4>Answer<\/h4>\r\n[latex]y=-\\frac{7}{8}x+\\frac{19}{4}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h2 id=\"video2\">Video: Find the Equation of a Line Given the Slope and a Point on the Line<\/h2>\r\nhttps:\/\/youtu.be\/URYnKqEctgc\r\n\r\n&nbsp;\r\n<h2 id=\"Find the Equation of a Line Given Two Points on the Line\">Find the Equation of a Line Given Two Points on the Line<\/h2>\r\nLet\u2019s suppose you don\u2019t know either the slope or the <i>y<\/i>-intercept, but you do know the location of two points on the line. It is more challenging, but you can find the equation of the line that would pass through those two points. You will again use slope-intercept form to help you.\r\n\r\nThe slope of a linear equation is always the same, no matter which two points you use to find the slope. Since you have two points, you can use those points to find the slope (<i>m<\/i>). Now you have the slope and a point on the line! You can now substitute values for <i>m<\/i>, <i>x<\/i>, and <i>y<\/i> into the equation [latex]y=mx+b[\/latex] and find <em>b<\/em>.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWrite the equation of the line that passes through the points [latex](2,1)[\/latex] and [latex](\u22121,\u22125)[\/latex].\r\n\r\n[reveal-answer q=\"333536\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"333536\"]\r\n\r\nFind the slope using the given points.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{1-(-5)}{2-(-1)}=\\frac{6}{3}=2[\/latex]<\/p>\r\nSubstitute the slope (<i>m<\/i>) into [latex]y=mx+b[\/latex].\r\n<p style=\"text-align: center;\">[latex]y=2x+b[\/latex]<\/p>\r\nSubstitute the coordinates of either point for <i>x <\/i>and <i>y<\/i>\u2013 this example uses\u00a0(2, 1).\r\n<p style=\"text-align: center;\">[latex]1=2(2)+b[\/latex]<\/p>\r\nSolve for <i>b<\/i>.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,1=4+b\\\\\u22123=b\\end{array}[\/latex]<\/p>\r\nRewrite [latex]y=mx+b[\/latex]\u00a0with [latex]m=2[\/latex] and [latex]b=-3[\/latex].\r\n<h4>Answer<\/h4>\r\n[latex]\\begin{array}{l}y=2x+\\left(-3\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{or}\\\\y=2x-3\\end{array}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nNotice that is doesn\u2019t matter which point you use when you substitute and solve for <i>b<\/i>\u2014you get the same result for <i>b<\/i> either way. In the example above, you substituted the coordinates of the point (2, 1) in the equation [latex]y=2x+b[\/latex]. Let\u2019s start with the same equation, [latex]y=2x+b[\/latex], but substitute in [latex](\u22121,\u22125)[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,y=2x+b\\\\-5=2\\left(-1\\right)+b\\\\-5=-2+b\\\\-3=b\\end{array}[\/latex]<\/p>\r\nThe final equation is the same: [latex]y=2x\u20133[\/latex].\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example (Advanced)<\/h3>\r\nWrite the equation of the line that passes through the points [latex](-4.6,6.45)[\/latex] and [latex](1.15,7.6)[\/latex].\r\n\r\n[reveal-answer q=\"347882\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"347882\"]\r\n\r\nFind the slope using the given points.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{7.6-6.45}{1.15-(-4.6)}=\\frac{1.15}{5.75}=0.2[\/latex]<\/p>\r\nSubstitute the slope (<i>m<\/i>) into [latex] \\displaystyle y=mx+b[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\displaystyle y=0.2x+b[\/latex]<\/p>\r\nSubstitute either point for <i>x <\/i>and <i>y\u2014<\/i>this example uses [latex](1.15,7.6)[\/latex]. Then solve for <i>b<\/i>.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{l}\\,\\,\\,\\,\\,\\,7.6\\,\\,=\\,\\,0.2(1.15)+b\\\\\\,\\,\\,\\,\\,\\,7.6\\,\\,=\\,\\,0.23+b\\\\\\,\\,\\,\\,\\,\\,7.6\\,\\,=\\,\\,0.23+b\\\\\\underline{-0.23\\,\\,\\,\\,-0.23}\\\\\\,\\,\\,\\,\\,7.37\\,=\\,\\,b\\end{array}[\/latex]<\/p>\r\nRewrite [latex] \\displaystyle y=mx+b[\/latex] with [latex]m=0.2[\/latex] and [latex]b=7.37[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\displaystyle y=0.2x+7.37[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nThe equation of the line that passes through the points [latex](-4.6,6.45)[\/latex] and [latex](1.15,7.6)[\/latex] is [latex]y=0.2x+7.37[\/latex].[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2 id=\"video3\">Video: Find the Equation of a Line Given Two Points on the Line<\/h2>\r\nhttps:\/\/youtu.be\/P1ex_a6iYDo\r\n\r\n&nbsp;\r\n<h2 id=\"Write the equations of parallel and perpendicular lines\">Write the equations of parallel and perpendicular lines<\/h2>\r\nThe relationships between slopes of parallel and perpendicular lines can be used to write equations of parallel and perpendicular lines.\r\n\r\nLet\u2019s start with an example involving parallel lines.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWrite the equation of a line that is parallel to the line [latex]x\u2013y=5[\/latex] and goes through the point [latex](\u22122,1)[\/latex].\r\n\r\n[reveal-answer q=\"763534\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"763534\"]\r\n\r\nRewrite the line you want to be parallel to into the\u00a0[latex]y=mx+b[\/latex] form, if needed.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x\u2013y=5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\u2212y=\u2212x+5\\\\y=x\u20135\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nIdentify the slope of the given line.\r\n\r\nIn the equation above, [latex]m=1[\/latex] and [latex]b=\u22125[\/latex].\r\n\r\nSince [latex]m=1[\/latex], the slope is 1.\r\n\r\nTo find the slope of a parallel line, use the same slope.\r\n\r\nThe slope of the parallel line is 1.\r\n\r\nUse the method for writing an equation from the slope and a point on the line. Substitute 1 for <i>m<\/i>, and the point [latex](\u22122,1)[\/latex] for <i>x<\/i> and <em>y<\/em>.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}y=mx+b\\\\1=1(\u22122)+b\\end{array}[\/latex]<\/p>\r\nSolve for <em>b<\/em>.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}1=\u22122+b\\\\3=b\\end{array}[\/latex]<\/p>\r\nWrite the equation using the new slope for <i>m<\/i> and the <i>b<\/i> you just found.\r\n<h4>Answer<\/h4>\r\n[latex]y=x+3[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h2 class=\"yt watch-title-container\"><span id=\"eow-title\" class=\"watch-title\" dir=\"ltr\" title=\"Determine the Equation of a Line Parallel to a Line in General Form\">Determine the Equation of a Line Parallel to Another Line Through a Given Point<\/span><\/h2>\r\nhttps:\/\/youtu.be\/TQKz2XHI09E\r\n<h2>Determine the Equation of a Line Perpendicular to Another Line Through a Given Point<\/h2>\r\nWhen you are working with perpendicular lines, you will usually be given one of the lines and an additional point. Remember that two non-vertical lines are perpendicular if the slope of one is the negative reciprocal of the slope of the other.\u00a0To find the slope of a perpendicular line, find the reciprocal, and then find the opposite of this reciprocal. \u00a0In other words, flip it and change the sign.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWrite the equation of a line that contains the point [latex](1,5)[\/latex] and is perpendicular to the line [latex]y=2x\u2013 6[\/latex].\r\n\r\n[reveal-answer q=\"604282\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"604282\"]\r\n\r\nIdentify the slope of the line you want to be perpendicular to.\r\n\r\nThe given line is written in [latex]y=mx+b[\/latex] form, with [latex]m=2[\/latex] and [latex]b=-6[\/latex]. The slope is 2.\r\n\r\nTo find the slope of a perpendicular line, find the reciprocal, [latex] \\displaystyle \\frac{1}{2}[\/latex], then the opposite, [latex] \\displaystyle -\\frac{1}{2}[\/latex].\r\n\r\nThe slope of the perpendicular line is [latex] \\displaystyle -\\frac{1}{2}[\/latex].\r\n\r\nUse the method for writing an equation from the slope and a point on the line. Substitute [latex] \\displaystyle -\\frac{1}{2}[\/latex] for <i>m<\/i>, and the point [latex](1,5)[\/latex] for <i>x<\/i> and <i>y<\/i>.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{l}y=mx+b\\\\5=-\\frac{1}{2}(1)+b\\end{array}[\/latex]<\/p>\r\nSolve for <i>b<\/i>.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{l}\\,\\,\\,5=-\\frac{1}{2}+b\\\\\\frac{11}{2}=b\\end{array}[\/latex]<\/p>\r\nWrite the equation using the new slope for <i>m<\/i> and the <i>b<\/i> you just found.\r\n<h4>Answer<\/h4>\r\n[latex]y=-\\frac{1}{2}x+\\frac{11}{2}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Determine the Equation of a Line Perpendicular to a Line in Slope-Intercept Form<\/h2>\r\nhttps:\/\/youtu.be\/QtvtzKjtowA\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWrite the equation of a line that is parallel to the line [latex]y=4[\/latex] through the point [latex](0,10)[\/latex].\r\n\r\n[reveal-answer q=\"426450\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"426450\"]\r\n\r\nRewrite the line into [latex]y=mx+b[\/latex]\u00a0form, if needed.\r\n\r\nYou may notice without doing this that [latex]y=4[\/latex]\u00a0is a horizontal line 4 units above the <i>x<\/i>-axis. Because it is horizontal, you know its slope is zero.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}y=4\\\\y=0x+4\\end{array}[\/latex]<\/p>\r\nIdentify the slope of the given line.\r\n\r\nIn the equation above, [latex]m=0[\/latex] and [latex]b=4[\/latex].\r\n\r\nSince [latex]m=0[\/latex], the slope is 0. This is a horizontal line.\r\n\r\nTo find the slope of a parallel line, use the same slope.\r\n\r\nThe slope of the parallel line is also 0.\r\n\r\nSince the parallel line will be a horizontal line, its form is\r\n<p style=\"text-align: center;\">[latex]y=\\text{a constant}[\/latex]<\/p>\r\nSince we want this new line to pass through the point [latex](0,10)[\/latex], we will need to write the equation of the new line as:\r\n<p style=\"text-align: center;\">[latex]y=10[\/latex]<\/p>\r\nThis line is parallel to [latex]y=4[\/latex]\u00a0and passes through [latex](0,10)[\/latex].\r\n<h4>Answer<\/h4>\r\n[latex]y=10[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWrite the equation of a line that is perpendicular to the line [latex]y=-3[\/latex] through the point [latex](-2,5)[\/latex].\r\n\r\n[reveal-answer q=\"426550\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"426550\"]\r\n\r\nIn the equation above, [latex]m=0[\/latex] and [latex]b=-3[\/latex].\r\n\r\nA perpendicular line will have a slope that is the negative reciprocal of the slope of\u00a0[latex]y=-3[\/latex], but\u00a0what does that mean in this case?\r\n\r\nThe reciprocal of 0 is [latex]\\frac{1}{0}[\/latex], but we know that dividing by 0 is undefined.\r\n\r\nThis means that we are looking for a line whose slope is undefined, and we also know that vertical lines have slopes that are undefined. This makes sense since we started with a horizontal line.\r\n\r\nThe form of a vertical line is [latex]x=\\text{a constant}[\/latex], where every <em>x<\/em>-value on the line is equal to some constant. \u00a0Since we are looking for a line that goes through the point [latex](-2,5)[\/latex], all of the <em>x<\/em>-values on this line must be [latex]-2[\/latex].\r\n\r\nThe equation of a line passing through [latex](-2,5)[\/latex] that is perpendicular to the horizontal line\u00a0[latex]y=-3[\/latex] is therefore,\r\n<p style=\"text-align: center;\">[latex]x=-2[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]x=-2[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h2 class=\"yt watch-title-container\"><span id=\"eow-title\" class=\"watch-title\" dir=\"ltr\" title=\"Ex: Find the Equation of a Perpendicular and Horizontal Line to a Horizontal Line\">Find the Equation of a Perpendicular and Horizontal Line to a Horizontal Line<\/span><\/h2>\r\nhttps:\/\/youtu.be\/Qpn3f3wMeIs\r\n<h2 class=\"yt watch-title-container\">\u00a0Interpret the <em>y<\/em>-intercept of a linear equation<\/h2>\r\nOften, when the line in question represents a set of data or observations, the <em>y<\/em>-intercept can be interpreted as a starting point. \u00a0We will continue to use the examples for house value in Mississippi and Hawaii and high school smokers to interpret the meaning of the <em>y<\/em>-intercept in those equations.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<strong>Recall the equations and data for house value:<\/strong>\r\n<p data-type=\"title\">Linear equations describing the change in median home values between 1950 and 2000 in Mississippi and Hawaii are as follows:<\/p>\r\n<p data-type=\"title\"><strong>Hawaii:\u00a0<\/strong> [latex]y = 3966x+74,400[\/latex]<\/p>\r\n<p data-type=\"title\"><strong>Mississippi:\u00a0\u00a0<\/strong>[latex]y = 924x+25,200[\/latex]<\/p>\r\n<p data-type=\"title\">The equations are based on the following dataset.<\/p>\r\n<p data-type=\"title\">x = the number of years since 1950, and y = the median value of a house in the given state.<\/p>\r\n\r\n<table id=\"Table_04_02_03\" summary=\"This table shows three rows and three columns. The first column is labeled: \u201cYear\u201d, the second: \u201cMississippi\u201d and the third: \u201cHawaii\u201d. The two year entries are: \u201c1950\u201d and \u201c2000\u201d. The two Mississippi entries are: \u201c$25,200\u201d and \u201c$71,400\u201d. The two Hawaii entries are: \u201c$74,400\u201d and \u201c$272,700\u201d.\">\r\n<thead>\r\n<tr>\r\n<th scope=\"col\" data-align=\"center\">Year (<em>x<\/em>)<\/th>\r\n<th scope=\"col\" data-align=\"center\">Mississippi House Value (<em>y<\/em>)<\/th>\r\n<th scope=\"col\" data-align=\"center\">Hawaii House Value (<em>y<\/em>)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"right\">0<\/td>\r\n<td data-align=\"right\">$25,200<\/td>\r\n<td data-align=\"right\">$74,400<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"right\">50<\/td>\r\n<td data-align=\"right\">$71,400<\/td>\r\n<td data-align=\"right\">$272,700<strong>\u00a0\u00a0<\/strong><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAnd the equations and data for high school smokers:\r\n<p data-type=\"title\">A linear equation describing the change in the number of high school students who smoke, in\u00a0a group of 100, between 2011 and 2015 is given as:<\/p>\r\n<p style=\"text-align: center;\" data-type=\"title\">\u00a0[latex]y = -1.75x+16[\/latex]<\/p>\r\n<p data-type=\"title\">And is based on the data from this table, provided by the Centers for Disease Control.<\/p>\r\n<p data-type=\"title\">x = the number of years since 2011, and y = the number of high school smokers per 100 students.<\/p>\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Year<\/td>\r\n<td>Number of \u00a0High School Students Smoking\u00a0Cigarettes (per 100)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0<\/td>\r\n<td>16<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td>9<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAlso recall that the equation of a line in slope-intercept form is as follows:\r\n<p style=\"text-align: center;\">[latex]y = mx + b[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,m\\,\\,\\,\\,=\\,\\,\\,\\text{slope}\\\\(x,y)=\\,\\,\\,\\text{a point on the line}\\\\\\,\\,\\,\\,\\,\\,\\,b\\,\\,\\,\\,=\\,\\,\\,\\text{the y value of the y-intercept}\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<p style=\"text-align: left;\">The examples that follow show how to interpret the y-intercept of the equations used to model house value and the number of high school smokers. Additionally, you will see how to use the equations to make predictions about house value and the number of smokers in future years.<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<p data-type=\"title\">Interpret the <em>y<\/em>-intercepts of the equations that represent the change in house value for Hawaii and Mississippi.<\/p>\r\n<p data-type=\"title\"><strong>Hawaii:\u00a0<\/strong> [latex]y = 3966x+74,400[\/latex]<\/p>\r\n<p data-type=\"title\"><strong>Mississippi:\u00a0\u00a0<\/strong>[latex]y = 924x+25,200[\/latex]<\/p>\r\nThe <em>y<\/em>-intercept of a two-variable linear equation can be found by substituting 0 in for x.\r\n<h4>Hawaii<\/h4>\r\n<p style=\"text-align: center;\">[latex]y = 3966x+74,400\\\\y = 3966(0)+74,400\\\\y = 74,400[\/latex]<\/p>\r\nThe <em>y<\/em>-intercept is a point, so we write it as (0, 74,400). \u00a0Remember that <em>y<\/em>-values represent dollars and <em>x<\/em> values represent years. \u00a0When the year is 0\u2014in this case 0\u00a0because that is the first date we have in the dataset\u2014the price of a house in Hawaii was $74,400.\r\n<h4>Mississippi<\/h4>\r\n<p style=\"text-align: center;\">[latex]y = 924x+25,200\\\\y = 924(0)+25,200\\\\y = 25,200[\/latex]<\/p>\r\nThe <em>y<\/em>-intercept is (0,\u00a025,200). \u00a0This means that in 1950 the value of a house in Mississippi was $25,200. Remember that <em>x<\/em> represents the number of years since 1950, so if [latex]x=0[\/latex] the year is 1950.\r\n\r\n<\/div>\r\n<h2 class=\"yt watch-title-container\"><\/h2>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<p data-type=\"title\">Interpret the y-intercept of the equation that represents the change in the number of high school students who smoke out of 100.<\/p>\r\nSubstitute 0 in for <em>x<\/em>.\r\n<p style=\"text-align: center;\">[latex]y = -1.75x+16\\\\y = -1.75(0)+16\\\\y = 16[\/latex]<\/p>\r\nThe y-intercept is [latex](0,16)[\/latex]. \u00a0The data starts at 2011, so we represent that year as 0. We can interpret the <em>y<\/em>-intercept as follows:\r\n\r\nIn the year 2011, 16 out of every 100 high school students smoked.\r\n\r\n<\/div>\r\n<p class=\"yt watch-title-container\"><span id=\"eow-title\" class=\"watch-title\" dir=\"ltr\" title=\"Interpret the Meaning of the y-intercept Given a Linear Equation\">In the following video you will see an example of how to interpret the y- intercept given a linear equation that represents a set of data.<\/span><\/p>\r\nhttps:\/\/youtu.be\/Yhtl28DRqfU\r\n<h2>Use a linear equation to make a prediction<\/h2>\r\nAnother useful outcome we gain from writing equations from data is the ability to make predictions about what may happen in the future. We will continue our analysis of the house price and high school smokers. In the following examples you will be shown how to predict future outcomes based on the linear equations that model current behavior.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse the equations for house value in Hawaii and Mississippi to predict house value in\u00a02035.\r\n\r\nWe are asked to find house value, <em>y<\/em>, when the year, <em>x<\/em>, is 2035. Since the equations we have represent house value increase since 1950, we have to be careful. We can't just plug in 2035 for <em>x<\/em>, because <em>x<\/em> represents the years since 1950.\r\n\r\nHow many years are between 1950 and 2035? [latex]2035 - 1950 = 85[\/latex]\r\n\r\nThis is our <em>x<\/em>-value.\r\n\r\nFor Hawaii:\r\n<p style=\"text-align: center;\">[latex]y = 3966x+74,400\\\\y = 3966(85)+74,400\\\\y = 337110+74,400 = 411,510[\/latex]<\/p>\r\nHoly cow! The average price for a house in Hawaii in 2035 is predicted to be $411,510 according to this model. See if you can find the <em>current<\/em> average value of a house in Hawaii. Does the model measure up?\r\n\r\nFor Mississippi:\r\n<p style=\"text-align: center;\">[latex]y = 924x+25,200\\\\y = 924(85)+25,200\\\\y = 78540+25,200 = 103,740[\/latex]<\/p>\r\nThe average price for a home in Mississippi in 2035 is predicted to be $103,740 according to the model.\u00a0See if you can find the <em>current<\/em> average value of a house in Mississippi. Does the model measure up?\r\n\r\n<\/div>\r\nIn the following video, you will see the example of how to make a prediction with the home value data.\r\n\r\nhttps:\/\/youtu.be\/Bw9XjDAl-K0\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse the equation for the number of high school smokers per 100 to predict the year when there will be 0 smokers per 100.\r\n<p style=\"text-align: center;\">[latex]y = -1.75x+16[\/latex]<\/p>\r\nThis question takes a little more thinking. \u00a0In terms of <em>x<\/em> and <em>y<\/em>, what does it mean to have 0 smokers? \u00a0Since <em>y<\/em> represents the number of smokers and <em>x<\/em> represent the year, we are being asked when y will be 0.\r\n\r\nSubstitute 0 for <em>y<\/em>.\r\n<p style=\"text-align: center;\">[latex]y = -1.75x+16[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]0 = -1.75x+16[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]-16 = -1.75x[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{-16}{-1.75} = x[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]x = 9.14[\/latex] years<\/p>\r\nAgain, like the last example,\u00a0<em>x<\/em> is representing the number of years since the start of the data\u2014which was 2011, based on the table:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Year<\/td>\r\n<td>Number of \u00a0High School Students Smoking\u00a0Cigarettes (per 100)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0<\/td>\r\n<td>16<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td>9<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo we are predicting that there will be no smokers in high school by [latex]2011+9.14=2020[\/latex]. How accurate do you think this model is? Do you think there will ever be 0 smokers in high school?\r\n\r\n<\/div>\r\nThe following video gives a thorough explanation of making a prediction given a linear equation.\r\n\r\nhttps:\/\/youtu.be\/5W0qq8saxO0\r\n<h2>Bringing it Together<\/h2>\r\nThe last example we will show will include many\u00a0of the concepts that we have been building up throughout this section. \u00a0We will interpret a word problem, write a linear equation from it, graph the equation, interpret the y-intercept and make a prediction. Hopefully this example will help you to make\u00a0connections between the concepts we have presented.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIt costs $600 to purchase an iphone, plus $55 per month for unlimited use and data.\r\n\r\nWrite a linear equation that represents the cost, y, \u00a0of owning and using the\u00a0iPhone for x amount of months. When you have written your equation, answer the following questions:\r\n<ol>\r\n \t<li>What is the total cost you\u2019ve paid after\u00a0owning and using your phone for 24 months?<\/li>\r\n \t<li>If you have spent\u00a0$2,580 since you purchased your phone, how many months have you used your phone?<\/li>\r\n<\/ol>\r\n[caption id=\"attachment_4649\" align=\"alignnone\" width=\"206\"]<img class=\" wp-image-4649\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/07175121\/Screen-Shot-2016-06-07-at-10.50.43-AM-300x220.png\" alt=\"5 iPhones laying next to each other\" width=\"206\" height=\"151\" \/> iPhone[\/caption]\r\n\r\n[reveal-answer q=\"282349\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"282349\"][\/hidden-answer]\r\n\r\n<strong>Read and Understand:<\/strong>\u00a0We need to write a linear equation that represents the cost of owning and using an iPhone for any number of months. \u00a0We are to use y to represent cost, and x to represent the number of months we have used the phone.\r\n\r\n<strong>Define and Translate:\u00a0<\/strong>We will use the slope-intercept form of a line, [latex]y=mx+b[\/latex], because we are given a starting cost and a monthly cost for use. \u00a0We will need to find the slope and the y-intercept.\r\n\r\nSlope: in this case we don't know two points, but we are given a rate in dollars for monthly use of the phone. \u00a0Our units are dollars per month because slope is [latex]\\frac{\\Delta{y}}{\\Delta{x}}[\/latex], and y is in dollars and x is in months. The slope will be [latex]\\frac{55\\text{ dollars }}{1\\text{ month }}[\/latex]. [latex]m=\\frac{55}{1}=55[\/latex]\r\n\r\nY-Intercept: the y-intercept is defined as a point [latex]\\left(0,b\\right)[\/latex]. \u00a0We want to know how much money we have spent, y, after 0 months. \u00a0We haven't paid for service yet, but we have paid $600 for the phone. The y-intercept in this case is called an initial cost. [latex]b=600[\/latex]\r\n\r\n<strong>Write and Solve:\u00a0<\/strong>Substitute the slope and intercept you defined into the slope=intercept equation.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}y=mx+b\\\\y=55x+600\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Now we will answer the following questions:<\/p>\r\n\r\n<ol>\r\n \t<li>What is the total cost you\u2019ve paid after\u00a0owning and using your phone for 24 months?<\/li>\r\n<\/ol>\r\nSince x represents the number of months you have used the phone, we can substitute x=24 into our equation.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}y=55x+600\\\\y=55\\left(24\\right)+600\\\\y=1320+600\\\\y=1920\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Y represents the cost after x number of months, so in this scenario, after 24 months, you have spent $1920 to own and use an iPhone.<\/p>\r\n\r\n<ol>\r\n \t<li>If you have spent\u00a0$2,580 since you purchased your phone, how many months have you used your phone?<\/li>\r\n<\/ol>\r\nWe know that y represents cost, and we are given a cost and asked to find the number of months related to having spent that much. We will substitute y=$2,580 into the equation, then use what we know about solving linear equations to isolate x:\r\n<p style=\"text-align: center;\">\u00a0[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y=55x+600\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2580=55x+600\\\\\\text{ subtract 600 from each side}\\,\\,\\,\\,\\,\\,\\,\\underline{-600}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-600}\\\\\\text{}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,1980=55x\\\\\\text{}\\\\\\text{ divide each side by 55 }\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{1980}{55}=\\frac{55x}{55}\\\\\\text{}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,36=x\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">If you have spent $2,580 then you have been using your iPhone for 36 months, or 3 years.<\/p>\r\n\r\n<\/div>\r\n<h3>Summary<\/h3>\r\nThe slope-intercept form of a linear equation is written as [latex]y=mx+b[\/latex], where <i>m<\/i> is the slope and <i>b<\/i> is the value of <i>y<\/i> at the <i>y<\/i>-intercept, which can be written as [latex](0,b)[\/latex]. When you know the slope and the <i>y<\/i>-intercept of a line you can use the slope-intercept form to immediately write the equation of that line. The slope-intercept form can also help you to write the equation of a line when you know the slope and a point on the line or when you know two points on the line.\r\n\r\nWhen lines in a plane are parallel (that is, they never cross), they have the same slope. When lines are perpendicular (that is, they cross at a 90\u00b0 angle), their slopes are opposite reciprocals of each other. The product of their slopes will be [latex]-1[\/latex], except in the case where one of the lines is vertical causing its slope to be undefined. You can use these relationships to find an equation of a line that goes through a particular point and is parallel or perpendicular to another line.\r\n<h2><\/h2>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Write the equation and draw the graph of a line using slope and y-intercept\n<ul>\n<li>Write the equation of a line using slope and y-intercept<\/li>\n<li>Rearrange a linear equation so it is in slope-intercept form.<\/li>\n<li>Graph a line using slope and y-intercept<\/li>\n<\/ul>\n<\/li>\n<li>Write and solve equations of lines using slope and a point on the line\n<ul>\n<li>Write the equation of a line given the slope and a point on the line.<\/li>\n<li>Identify which parts of a linear equation are given and which parts need to be solved for using algebra<\/li>\n<\/ul>\n<\/li>\n<li>Write and solve equations of lines using two points on the line\n<ul>\n<li>Write the equation of a line given two points on the line<\/li>\n<li>Identify which parts of a linear equation are given and which parts need to be solved for using algebra.<\/li>\n<\/ul>\n<\/li>\n<li>Write equations of parallel and perpendicular lines\n<ul>\n<li>Find a line that is parallel to another line given a point<\/li>\n<li>Find a line that is perpendicular to another line given a point<\/li>\n<\/ul>\n<\/li>\n<li>Interpret the y-intercept of a linear equation and use that equation to make predictions\n<ul>\n<li>Interpret the <em>y<\/em>-intercept of a linear equation<\/li>\n<li>Use a linear equation to make a prediction<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<p>When graphing a line we found one method we could use is to make a table of values. However, if we can identify some properties of the line, we may be able to make a graph much quicker and easier. One such method is finding the slope and the <em>y<\/em>-intercept of the equation. The slope can be represented by m and the <em>y<\/em>-intercept, where it crosses the axis and [latex]x=0[\/latex], can be represented by [latex](0,b)[\/latex] where <em>b<\/em> is the value where the graph crosses the vertical <em>y<\/em>-axis. Any other point on the line can be represented by [latex](x,y)[\/latex].<\/p>\n<p>In the equation,<\/p>\n<p style=\"text-align: center;\">[latex]y = mx + b[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,m\\,\\,\\,\\,=\\,\\,\\,\\text{slope}\\\\(x,y)=\\,\\,\\,\\text{a point on the line}\\\\\\,\\,\\,\\,\\,\\,\\,b\\,\\,\\,\\,=\\,\\,\\,\\text{the y value of the y-intercept}\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">This formula is known as the slope-intercept equation.\u00a0If we know the slope and the <em>y<\/em>-intercept we can easily find the equation that represents the line<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Write the equation of the line that has a slope of [latex]\\displaystyle \\frac{1}{2}[\/latex] and a <i>y<\/i>-intercept of [latex]\u22125[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q624715\">Show Solution<\/span><\/p>\n<div id=\"q624715\" class=\"hidden-answer\" style=\"display: none\">Substitute the slope (<i>m<\/i>) into [latex]y=mx+b[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle y=\\frac{1}{2}x+b[\/latex]<\/p>\n<p>Substitute the <i>y<\/i>-intercept (<i>b<\/i>) into the equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle y=\\frac{1}{2}x-5[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]y=\\frac{1}{2}x-5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>We can also find the equation by looking at a graph and finding the slope and <em>y<\/em>-intercept.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Write the equation of the line in the graph by identifying the slope and <em>y<\/em>-intercept.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-3198 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/26035543\/SVG_Grapher-300x297.png\" alt=\"SVG_Grapher\" width=\"300\" height=\"297\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q96446\">Show Solution<\/span><\/p>\n<div id=\"q96446\" class=\"hidden-answer\" style=\"display: none\">Identify the point where the graph crosses the y-axis [latex](0,3)[\/latex]. This means the <em>y<\/em>-intercept is 3.<\/p>\n<p>Identify one other point and draw a slope triangle to find the slope.<\/p>\n<p>The slope is [latex]\\frac{-2}{3}[\/latex]<\/p>\n<p>Substitute the slope and <em>y<\/em> value of the intercept into the slope-intercept equation.<\/p>\n<p style=\"text-align: center;\">[latex]y=mx+b\\\\y=\\frac{-2}{3}x+b\\\\y=\\frac{-2}{3}x+3[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]y=\\frac{-2}{3}x+3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>We can also move the opposite direction, using the equation identify the slope\u00a0and <em>y<\/em>-intercept and graph the equation from this information. However, it will be\u00a0important for the equation to first be in slope intercept form. If it is not, we will\u00a0have to solve it for <em>y<\/em> so we can identify the slope and the <em>y<\/em>-intercept.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Write the following equation in slope-intercept form.<\/p>\n<p style=\"text-align: center;\">[latex]2x+4y=6[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q373034\">Show Solution<\/span><\/p>\n<div id=\"q373034\" class=\"hidden-answer\" style=\"display: none\">We need to solve for <em>y<\/em>. Start by subtracting [latex]2[\/latex] from both sides.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2x\\,\\,\\,+\\,\\,\\,4y\\,\\,\\,=\\,\\,\\,6\\\\-2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-2x\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">It helps to place the <em>x<\/em> term first on the right hand side. Notice how we keep the 6 positive by placing an addition sign in front.<\/p>\n<p style=\"text-align: center;\">[latex]4y=-2x+6[\/latex]<\/p>\n<p style=\"text-align: left;\">Divide each term by 4 to isolate the <em>y<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{4y}{4}=\\frac{-2x}{4}+\\frac{6}{4}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]y=\\frac{-2x}{4}+\\frac{6}{4}[\/latex]<\/p>\n<p style=\"text-align: left;\">Reduce the fractions<\/p>\n<p style=\"text-align: center;\">[latex]y=-\\frac{1}{2}x+\\frac{3}{2}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]y=-\\frac{1}{2}x+\\frac{3}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Once we have an equation in slope-intercept form we can graph it by first plotting\u00a0the <em>y<\/em>-intercept, then using the slope, find a second point and connecting the dots.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Graph [latex]y=\\frac{1}{2}x-4[\/latex] using the slope-intercept equation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q420487\">Show Solution<\/span><\/p>\n<div id=\"q420487\" class=\"hidden-answer\" style=\"display: none\">First, plot the <em>y<\/em>-intercept.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3202 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/26042734\/SVG_Grapher2-300x294.png\" alt=\"The y-intercept plotted at negative 4 on the y axis.\" width=\"300\" height=\"294\" \/><\/p>\n<p>Now use the slope to count up or down and over left or right to the next point. This slope is [latex]\\frac{1}{2}[\/latex], so you can count up one and right two\u2014both positive because both parts of the slope are positive.<\/p>\n<p>Connect the dots.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3203 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/26043304\/SVG_Grapher3-300x289.png\" alt=\"A line crosses through negative 4 on the y-axis and has a slope of 1\/2.\" width=\"300\" height=\"289\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Slope-Intercept Form of a Line<\/h2>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Find the Equation of a Line in Slope-Intercept Form of a Line (Basic)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/GIn7vbB5AYo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<h2 id=\"Find the Equation of a Line Given the Slope and a Point on the Line\">Find the Equation of a Line Given the Slope and a Point on the Line<\/h2>\n<p>Using the slope-intercept equation of a line is possible when you know both the slope (<i>m<\/i>) and the <i>y<\/i>-intercept (<i>b<\/i>), but what if you know the slope and just any point on the line, not specifically the <i>y<\/i>-intercept? Can you still write the equation? The answer is <i>yes<\/i>, but you will need to put in a little more thought and work than you did previously.<\/p>\n<p>Recall that a point is an (<i>x<\/i>, <i>y<\/i>) coordinate pair and that all points on the line will satisfy the linear equation. So, if you have a point on the line, it must be a solution to the equation. Although you don\u2019t know the exact equation yet, you know that you can express the line in slope-intercept form, [latex]y=mx+b[\/latex].<\/p>\n<p>You do know the slope (<i>m<\/i>), but you just don\u2019t know the value of the <i>y<\/i>-intercept (<i>b<\/i>). Since point (<i>x<\/i>, <i>y<\/i>) is a solution to the equation, you can substitute its coordinates for <i>x<\/i> and <i>y<\/i> in [latex]y=mx+b[\/latex]\u00a0and solve to find <i>b<\/i>!<\/p>\n<p>This may seem a bit confusing with all the variables, but an example with an actual slope and a point will help to clarify.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Write the equation of the line that has a slope of 3 and contains the point [latex](1,4)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q161353\">Show Solution<\/span><\/p>\n<div id=\"q161353\" class=\"hidden-answer\" style=\"display: none\">\n<p>Substitute the slope (<i>m<\/i>) into\u00a0[latex]y=mx+b[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]y=3x+b[\/latex]<\/p>\n<p>Substitute the point [latex](1,4)[\/latex] for <i>x <\/i>and <i>y.<\/i><\/p>\n<p style=\"text-align: center;\">[latex]4=3\\left(1\\right)+b[\/latex]<\/p>\n<p>Solve for <i>b.<\/i><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}4=3+b\\\\1=b\\end{array}[\/latex]<\/p>\n<p>Rewrite [latex]y=mx+b[\/latex]\u00a0with [latex]m=3[\/latex]\u00a0and [latex]b=1[\/latex].<\/p>\n<h4>Answer<\/h4>\n<p>[latex]y=3x+1[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>To confirm our algebra, you can check by graphing the equation [latex]y=3x+1[\/latex]. The equation checks because when graphed it passes through the point [latex](1,4)[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064327\/image045.jpg\" alt=\"An uphill line passes through the y-intercept of (0,1) and the point (1,4). The rise is 3 and the run is 1.\" width=\"348\" height=\"349\" \/><\/p>\n<p>If you know the slope of a line and a point on the line, you can draw a graph. Using an equation in the point-slope form allows you to identify the slope and a point. Consider the equation [latex]\\displaystyle y=-3x-1[\/latex]. The <em>y<\/em>-intercept is the point on the line where it passes through the <em>y<\/em>-axis. What is the value of <em>x<\/em> at this point?<\/p>\n<div class=\"textbox shaded\">Reminder: All <em>y<\/em>-intercepts are points in the form [latex](0,y)[\/latex]. \u00a0The <em>x<\/em> value of any <em>y<\/em>-intercept is <em>always<\/em>\u00a0zero.<\/div>\n<p>Therefore, you can tell from this equation that the <i>y<\/i>-intercept is at [latex](0,\u22121)[\/latex], check this by replacing <em>x<\/em> with 0 and solving for <em>y<\/em>. To graph the line, start by plotting that point, [latex](0,\u22121)[\/latex], on a graph.<\/p>\n<p>You can also tell from the equation that the slope of this line is [latex]\u22123[\/latex]. So start at [latex](0,\u22121)[\/latex] and count up 3 and over [latex]\u22121[\/latex] (1 unit in the negative direction, left) and plot a second point. (You could also have gone down 3 and over 1.) Then draw a line through both points, and there it is, the graph of [latex]\\displaystyle y=-3x-1[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064324\/image044.jpg\" alt=\"A downhill line passes through the point (-1,2) and the y-intercept (0,-1). The rise is 3 and the run is -1.\" width=\"325\" height=\"326\" \/><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example (Advanced)<\/h3>\n<p>Write the equation of the line that has a slope of [latex]-\\frac{7}{8}[\/latex]\u00a0and contains the point [latex]\\left(4,\\frac{5}{4}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q31452\">Show Solution<\/span><\/p>\n<div id=\"q31452\" class=\"hidden-answer\" style=\"display: none\">\n<p>Substitute the slope (<i>m<\/i>) into [latex]y=mx+b[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}y=mx+b\\\\\\\\y=-\\frac{7}{8}x+b\\end{array}[\/latex]<\/p>\n<p>Substitute the point [latex]\\left(4,\\frac{5}{4}\\right)[\/latex]\u00a0for <i>x <\/i>and <i>y.<\/i><\/p>\n<p style=\"text-align: center;\">[latex]\\frac{5}{4}=-\\frac{7}{8}\\left(4\\right)+b[\/latex]<\/p>\n<p>Solve for <i>b.<\/i><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{5}{4}=-\\frac{28}{8}+b\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{5}{4}=-\\frac{14}{4}+b\\\\\\\\\\frac{5}{4}+\\frac{14}{4}=-\\frac{14}{4}+\\frac{14}{4}+b\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{19}{4}=b\\end{array}[\/latex]<\/p>\n<p>Rewrite [latex]y=mx+b[\/latex] with [latex]\\displaystyle m=-\\frac{7}{8}[\/latex] and [latex]\\displaystyle b=\\frac{19}{4}[\/latex].<\/p>\n<h4>Answer<\/h4>\n<p>[latex]y=-\\frac{7}{8}x+\\frac{19}{4}[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<h2 id=\"video2\">Video: Find the Equation of a Line Given the Slope and a Point on the Line<\/h2>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex: Determine a Linear Equation Given Slope and a Point (Slope-Intercept Form) (09x-32)\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/URYnKqEctgc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<h2 id=\"Find the Equation of a Line Given Two Points on the Line\">Find the Equation of a Line Given Two Points on the Line<\/h2>\n<p>Let\u2019s suppose you don\u2019t know either the slope or the <i>y<\/i>-intercept, but you do know the location of two points on the line. It is more challenging, but you can find the equation of the line that would pass through those two points. You will again use slope-intercept form to help you.<\/p>\n<p>The slope of a linear equation is always the same, no matter which two points you use to find the slope. Since you have two points, you can use those points to find the slope (<i>m<\/i>). Now you have the slope and a point on the line! You can now substitute values for <i>m<\/i>, <i>x<\/i>, and <i>y<\/i> into the equation [latex]y=mx+b[\/latex] and find <em>b<\/em>.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Write the equation of the line that passes through the points [latex](2,1)[\/latex] and [latex](\u22121,\u22125)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q333536\">Show Solution<\/span><\/p>\n<div id=\"q333536\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find the slope using the given points.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{1-(-5)}{2-(-1)}=\\frac{6}{3}=2[\/latex]<\/p>\n<p>Substitute the slope (<i>m<\/i>) into [latex]y=mx+b[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]y=2x+b[\/latex]<\/p>\n<p>Substitute the coordinates of either point for <i>x <\/i>and <i>y<\/i>\u2013 this example uses\u00a0(2, 1).<\/p>\n<p style=\"text-align: center;\">[latex]1=2(2)+b[\/latex]<\/p>\n<p>Solve for <i>b<\/i>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,1=4+b\\\\\u22123=b\\end{array}[\/latex]<\/p>\n<p>Rewrite [latex]y=mx+b[\/latex]\u00a0with [latex]m=2[\/latex] and [latex]b=-3[\/latex].<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\begin{array}{l}y=2x+\\left(-3\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{or}\\\\y=2x-3\\end{array}[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>Notice that is doesn\u2019t matter which point you use when you substitute and solve for <i>b<\/i>\u2014you get the same result for <i>b<\/i> either way. In the example above, you substituted the coordinates of the point (2, 1) in the equation [latex]y=2x+b[\/latex]. Let\u2019s start with the same equation, [latex]y=2x+b[\/latex], but substitute in [latex](\u22121,\u22125)[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,y=2x+b\\\\-5=2\\left(-1\\right)+b\\\\-5=-2+b\\\\-3=b\\end{array}[\/latex]<\/p>\n<p>The final equation is the same: [latex]y=2x\u20133[\/latex].<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example (Advanced)<\/h3>\n<p>Write the equation of the line that passes through the points [latex](-4.6,6.45)[\/latex] and [latex](1.15,7.6)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q347882\">Show Solution<\/span><\/p>\n<div id=\"q347882\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find the slope using the given points.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{7.6-6.45}{1.15-(-4.6)}=\\frac{1.15}{5.75}=0.2[\/latex]<\/p>\n<p>Substitute the slope (<i>m<\/i>) into [latex]\\displaystyle y=mx+b[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle y=0.2x+b[\/latex]<\/p>\n<p>Substitute either point for <i>x <\/i>and <i>y\u2014<\/i>this example uses [latex](1.15,7.6)[\/latex]. Then solve for <i>b<\/i>.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{l}\\,\\,\\,\\,\\,\\,7.6\\,\\,=\\,\\,0.2(1.15)+b\\\\\\,\\,\\,\\,\\,\\,7.6\\,\\,=\\,\\,0.23+b\\\\\\,\\,\\,\\,\\,\\,7.6\\,\\,=\\,\\,0.23+b\\\\\\underline{-0.23\\,\\,\\,\\,-0.23}\\\\\\,\\,\\,\\,\\,7.37\\,=\\,\\,b\\end{array}[\/latex]<\/p>\n<p>Rewrite [latex]\\displaystyle y=mx+b[\/latex] with [latex]m=0.2[\/latex] and [latex]b=7.37[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle y=0.2x+7.37[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>The equation of the line that passes through the points [latex](-4.6,6.45)[\/latex] and [latex](1.15,7.6)[\/latex] is [latex]y=0.2x+7.37[\/latex].<\/p><\/div>\n<\/div>\n<\/div>\n<h2 id=\"video3\">Video: Find the Equation of a Line Given Two Points on the Line<\/h2>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex 1:  Find the Equation of a Line in Slope Intercept Form Given Two Points\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/P1ex_a6iYDo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<h2 id=\"Write the equations of parallel and perpendicular lines\">Write the equations of parallel and perpendicular lines<\/h2>\n<p>The relationships between slopes of parallel and perpendicular lines can be used to write equations of parallel and perpendicular lines.<\/p>\n<p>Let\u2019s start with an example involving parallel lines.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Write the equation of a line that is parallel to the line [latex]x\u2013y=5[\/latex] and goes through the point [latex](\u22122,1)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q763534\">Show Solution<\/span><\/p>\n<div id=\"q763534\" class=\"hidden-answer\" style=\"display: none\">\n<p>Rewrite the line you want to be parallel to into the\u00a0[latex]y=mx+b[\/latex] form, if needed.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x\u2013y=5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\u2212y=\u2212x+5\\\\y=x\u20135\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Identify the slope of the given line.<\/p>\n<p>In the equation above, [latex]m=1[\/latex] and [latex]b=\u22125[\/latex].<\/p>\n<p>Since [latex]m=1[\/latex], the slope is 1.<\/p>\n<p>To find the slope of a parallel line, use the same slope.<\/p>\n<p>The slope of the parallel line is 1.<\/p>\n<p>Use the method for writing an equation from the slope and a point on the line. Substitute 1 for <i>m<\/i>, and the point [latex](\u22122,1)[\/latex] for <i>x<\/i> and <em>y<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}y=mx+b\\\\1=1(\u22122)+b\\end{array}[\/latex]<\/p>\n<p>Solve for <em>b<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}1=\u22122+b\\\\3=b\\end{array}[\/latex]<\/p>\n<p>Write the equation using the new slope for <i>m<\/i> and the <i>b<\/i> you just found.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]y=x+3[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<h2 class=\"yt watch-title-container\"><span id=\"eow-title\" class=\"watch-title\" dir=\"ltr\" title=\"Determine the Equation of a Line Parallel to a Line in General Form\">Determine the Equation of a Line Parallel to Another Line Through a Given Point<\/span><\/h2>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Determine the Equation of a Line Parallel to a Line in General Form\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/TQKz2XHI09E?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Determine the Equation of a Line Perpendicular to Another Line Through a Given Point<\/h2>\n<p>When you are working with perpendicular lines, you will usually be given one of the lines and an additional point. Remember that two non-vertical lines are perpendicular if the slope of one is the negative reciprocal of the slope of the other.\u00a0To find the slope of a perpendicular line, find the reciprocal, and then find the opposite of this reciprocal. \u00a0In other words, flip it and change the sign.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Write the equation of a line that contains the point [latex](1,5)[\/latex] and is perpendicular to the line [latex]y=2x\u2013 6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q604282\">Show Solution<\/span><\/p>\n<div id=\"q604282\" class=\"hidden-answer\" style=\"display: none\">\n<p>Identify the slope of the line you want to be perpendicular to.<\/p>\n<p>The given line is written in [latex]y=mx+b[\/latex] form, with [latex]m=2[\/latex] and [latex]b=-6[\/latex]. The slope is 2.<\/p>\n<p>To find the slope of a perpendicular line, find the reciprocal, [latex]\\displaystyle \\frac{1}{2}[\/latex], then the opposite, [latex]\\displaystyle -\\frac{1}{2}[\/latex].<\/p>\n<p>The slope of the perpendicular line is [latex]\\displaystyle -\\frac{1}{2}[\/latex].<\/p>\n<p>Use the method for writing an equation from the slope and a point on the line. Substitute [latex]\\displaystyle -\\frac{1}{2}[\/latex] for <i>m<\/i>, and the point [latex](1,5)[\/latex] for <i>x<\/i> and <i>y<\/i>.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{l}y=mx+b\\\\5=-\\frac{1}{2}(1)+b\\end{array}[\/latex]<\/p>\n<p>Solve for <i>b<\/i>.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{l}\\,\\,\\,5=-\\frac{1}{2}+b\\\\\\frac{11}{2}=b\\end{array}[\/latex]<\/p>\n<p>Write the equation using the new slope for <i>m<\/i> and the <i>b<\/i> you just found.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]y=-\\frac{1}{2}x+\\frac{11}{2}[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<h2>Determine the Equation of a Line Perpendicular to a Line in Slope-Intercept Form<\/h2>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Determine the Equation of a Line Perpendicular to a Line in Slope-Intercept Form\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/QtvtzKjtowA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Write the equation of a line that is parallel to the line [latex]y=4[\/latex] through the point [latex](0,10)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q426450\">Show Solution<\/span><\/p>\n<div id=\"q426450\" class=\"hidden-answer\" style=\"display: none\">\n<p>Rewrite the line into [latex]y=mx+b[\/latex]\u00a0form, if needed.<\/p>\n<p>You may notice without doing this that [latex]y=4[\/latex]\u00a0is a horizontal line 4 units above the <i>x<\/i>-axis. Because it is horizontal, you know its slope is zero.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}y=4\\\\y=0x+4\\end{array}[\/latex]<\/p>\n<p>Identify the slope of the given line.<\/p>\n<p>In the equation above, [latex]m=0[\/latex] and [latex]b=4[\/latex].<\/p>\n<p>Since [latex]m=0[\/latex], the slope is 0. This is a horizontal line.<\/p>\n<p>To find the slope of a parallel line, use the same slope.<\/p>\n<p>The slope of the parallel line is also 0.<\/p>\n<p>Since the parallel line will be a horizontal line, its form is<\/p>\n<p style=\"text-align: center;\">[latex]y=\\text{a constant}[\/latex]<\/p>\n<p>Since we want this new line to pass through the point [latex](0,10)[\/latex], we will need to write the equation of the new line as:<\/p>\n<p style=\"text-align: center;\">[latex]y=10[\/latex]<\/p>\n<p>This line is parallel to [latex]y=4[\/latex]\u00a0and passes through [latex](0,10)[\/latex].<\/p>\n<h4>Answer<\/h4>\n<p>[latex]y=10[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Write the equation of a line that is perpendicular to the line [latex]y=-3[\/latex] through the point [latex](-2,5)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q426550\">Show Solution<\/span><\/p>\n<div id=\"q426550\" class=\"hidden-answer\" style=\"display: none\">\n<p>In the equation above, [latex]m=0[\/latex] and [latex]b=-3[\/latex].<\/p>\n<p>A perpendicular line will have a slope that is the negative reciprocal of the slope of\u00a0[latex]y=-3[\/latex], but\u00a0what does that mean in this case?<\/p>\n<p>The reciprocal of 0 is [latex]\\frac{1}{0}[\/latex], but we know that dividing by 0 is undefined.<\/p>\n<p>This means that we are looking for a line whose slope is undefined, and we also know that vertical lines have slopes that are undefined. This makes sense since we started with a horizontal line.<\/p>\n<p>The form of a vertical line is [latex]x=\\text{a constant}[\/latex], where every <em>x<\/em>-value on the line is equal to some constant. \u00a0Since we are looking for a line that goes through the point [latex](-2,5)[\/latex], all of the <em>x<\/em>-values on this line must be [latex]-2[\/latex].<\/p>\n<p>The equation of a line passing through [latex](-2,5)[\/latex] that is perpendicular to the horizontal line\u00a0[latex]y=-3[\/latex] is therefore,<\/p>\n<p style=\"text-align: center;\">[latex]x=-2[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]x=-2[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<h2 class=\"yt watch-title-container\"><span id=\"eow-title\" class=\"watch-title\" dir=\"ltr\" title=\"Ex: Find the Equation of a Perpendicular and Horizontal Line to a Horizontal Line\">Find the Equation of a Perpendicular and Horizontal Line to a Horizontal Line<\/span><\/h2>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Ex: Find the Equation of a Perpendicular and Horizontal Line to a Horizontal Line\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Qpn3f3wMeIs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 class=\"yt watch-title-container\">\u00a0Interpret the <em>y<\/em>-intercept of a linear equation<\/h2>\n<p>Often, when the line in question represents a set of data or observations, the <em>y<\/em>-intercept can be interpreted as a starting point. \u00a0We will continue to use the examples for house value in Mississippi and Hawaii and high school smokers to interpret the meaning of the <em>y<\/em>-intercept in those equations.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p><strong>Recall the equations and data for house value:<\/strong><\/p>\n<p data-type=\"title\">Linear equations describing the change in median home values between 1950 and 2000 in Mississippi and Hawaii are as follows:<\/p>\n<p data-type=\"title\"><strong>Hawaii:\u00a0<\/strong> [latex]y = 3966x+74,400[\/latex]<\/p>\n<p data-type=\"title\"><strong>Mississippi:\u00a0\u00a0<\/strong>[latex]y = 924x+25,200[\/latex]<\/p>\n<p data-type=\"title\">The equations are based on the following dataset.<\/p>\n<p data-type=\"title\">x = the number of years since 1950, and y = the median value of a house in the given state.<\/p>\n<table id=\"Table_04_02_03\" summary=\"This table shows three rows and three columns. The first column is labeled: \u201cYear\u201d, the second: \u201cMississippi\u201d and the third: \u201cHawaii\u201d. The two year entries are: \u201c1950\u201d and \u201c2000\u201d. The two Mississippi entries are: \u201c$25,200\u201d and \u201c$71,400\u201d. The two Hawaii entries are: \u201c$74,400\u201d and \u201c$272,700\u201d.\">\n<thead>\n<tr>\n<th scope=\"col\" data-align=\"center\">Year (<em>x<\/em>)<\/th>\n<th scope=\"col\" data-align=\"center\">Mississippi House Value (<em>y<\/em>)<\/th>\n<th scope=\"col\" data-align=\"center\">Hawaii House Value (<em>y<\/em>)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-align=\"right\">0<\/td>\n<td data-align=\"right\">$25,200<\/td>\n<td data-align=\"right\">$74,400<\/td>\n<\/tr>\n<tr>\n<td data-align=\"right\">50<\/td>\n<td data-align=\"right\">$71,400<\/td>\n<td data-align=\"right\">$272,700<strong>\u00a0\u00a0<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>And the equations and data for high school smokers:<\/p>\n<p data-type=\"title\">A linear equation describing the change in the number of high school students who smoke, in\u00a0a group of 100, between 2011 and 2015 is given as:<\/p>\n<p style=\"text-align: center;\" data-type=\"title\">\u00a0[latex]y = -1.75x+16[\/latex]<\/p>\n<p data-type=\"title\">And is based on the data from this table, provided by the Centers for Disease Control.<\/p>\n<p data-type=\"title\">x = the number of years since 2011, and y = the number of high school smokers per 100 students.<\/p>\n<table>\n<tbody>\n<tr>\n<td>Year<\/td>\n<td>Number of \u00a0High School Students Smoking\u00a0Cigarettes (per 100)<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>16<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Also recall that the equation of a line in slope-intercept form is as follows:<\/p>\n<p style=\"text-align: center;\">[latex]y = mx + b[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,m\\,\\,\\,\\,=\\,\\,\\,\\text{slope}\\\\(x,y)=\\,\\,\\,\\text{a point on the line}\\\\\\,\\,\\,\\,\\,\\,\\,b\\,\\,\\,\\,=\\,\\,\\,\\text{the y value of the y-intercept}\\end{array}[\/latex]<\/p>\n<\/div>\n<p style=\"text-align: left;\">The examples that follow show how to interpret the y-intercept of the equations used to model house value and the number of high school smokers. Additionally, you will see how to use the equations to make predictions about house value and the number of smokers in future years.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p data-type=\"title\">Interpret the <em>y<\/em>-intercepts of the equations that represent the change in house value for Hawaii and Mississippi.<\/p>\n<p data-type=\"title\"><strong>Hawaii:\u00a0<\/strong> [latex]y = 3966x+74,400[\/latex]<\/p>\n<p data-type=\"title\"><strong>Mississippi:\u00a0\u00a0<\/strong>[latex]y = 924x+25,200[\/latex]<\/p>\n<p>The <em>y<\/em>-intercept of a two-variable linear equation can be found by substituting 0 in for x.<\/p>\n<h4>Hawaii<\/h4>\n<p style=\"text-align: center;\">[latex]y = 3966x+74,400\\\\y = 3966(0)+74,400\\\\y = 74,400[\/latex]<\/p>\n<p>The <em>y<\/em>-intercept is a point, so we write it as (0, 74,400). \u00a0Remember that <em>y<\/em>-values represent dollars and <em>x<\/em> values represent years. \u00a0When the year is 0\u2014in this case 0\u00a0because that is the first date we have in the dataset\u2014the price of a house in Hawaii was $74,400.<\/p>\n<h4>Mississippi<\/h4>\n<p style=\"text-align: center;\">[latex]y = 924x+25,200\\\\y = 924(0)+25,200\\\\y = 25,200[\/latex]<\/p>\n<p>The <em>y<\/em>-intercept is (0,\u00a025,200). \u00a0This means that in 1950 the value of a house in Mississippi was $25,200. Remember that <em>x<\/em> represents the number of years since 1950, so if [latex]x=0[\/latex] the year is 1950.<\/p>\n<\/div>\n<h2 class=\"yt watch-title-container\"><\/h2>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p data-type=\"title\">Interpret the y-intercept of the equation that represents the change in the number of high school students who smoke out of 100.<\/p>\n<p>Substitute 0 in for <em>x<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]y = -1.75x+16\\\\y = -1.75(0)+16\\\\y = 16[\/latex]<\/p>\n<p>The y-intercept is [latex](0,16)[\/latex]. \u00a0The data starts at 2011, so we represent that year as 0. We can interpret the <em>y<\/em>-intercept as follows:<\/p>\n<p>In the year 2011, 16 out of every 100 high school students smoked.<\/p>\n<\/div>\n<p class=\"yt watch-title-container\"><span id=\"eow-title\" class=\"watch-title\" dir=\"ltr\" title=\"Interpret the Meaning of the y-intercept Given a Linear Equation\">In the following video you will see an example of how to interpret the y- intercept given a linear equation that represents a set of data.<\/span><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-7\" title=\"Interpret the Meaning of the y-intercept Given a Linear Equation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Yhtl28DRqfU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Use a linear equation to make a prediction<\/h2>\n<p>Another useful outcome we gain from writing equations from data is the ability to make predictions about what may happen in the future. We will continue our analysis of the house price and high school smokers. In the following examples you will be shown how to predict future outcomes based on the linear equations that model current behavior.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use the equations for house value in Hawaii and Mississippi to predict house value in\u00a02035.<\/p>\n<p>We are asked to find house value, <em>y<\/em>, when the year, <em>x<\/em>, is 2035. Since the equations we have represent house value increase since 1950, we have to be careful. We can&#8217;t just plug in 2035 for <em>x<\/em>, because <em>x<\/em> represents the years since 1950.<\/p>\n<p>How many years are between 1950 and 2035? [latex]2035 - 1950 = 85[\/latex]<\/p>\n<p>This is our <em>x<\/em>-value.<\/p>\n<p>For Hawaii:<\/p>\n<p style=\"text-align: center;\">[latex]y = 3966x+74,400\\\\y = 3966(85)+74,400\\\\y = 337110+74,400 = 411,510[\/latex]<\/p>\n<p>Holy cow! The average price for a house in Hawaii in 2035 is predicted to be $411,510 according to this model. See if you can find the <em>current<\/em> average value of a house in Hawaii. Does the model measure up?<\/p>\n<p>For Mississippi:<\/p>\n<p style=\"text-align: center;\">[latex]y = 924x+25,200\\\\y = 924(85)+25,200\\\\y = 78540+25,200 = 103,740[\/latex]<\/p>\n<p>The average price for a home in Mississippi in 2035 is predicted to be $103,740 according to the model.\u00a0See if you can find the <em>current<\/em> average value of a house in Mississippi. Does the model measure up?<\/p>\n<\/div>\n<p>In the following video, you will see the example of how to make a prediction with the home value data.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-8\" title=\"Make a Prediction Using a Linear Equation - Home Value\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Bw9XjDAl-K0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use the equation for the number of high school smokers per 100 to predict the year when there will be 0 smokers per 100.<\/p>\n<p style=\"text-align: center;\">[latex]y = -1.75x+16[\/latex]<\/p>\n<p>This question takes a little more thinking. \u00a0In terms of <em>x<\/em> and <em>y<\/em>, what does it mean to have 0 smokers? \u00a0Since <em>y<\/em> represents the number of smokers and <em>x<\/em> represent the year, we are being asked when y will be 0.<\/p>\n<p>Substitute 0 for <em>y<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]y = -1.75x+16[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]0 = -1.75x+16[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]-16 = -1.75x[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{-16}{-1.75} = x[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x = 9.14[\/latex] years<\/p>\n<p>Again, like the last example,\u00a0<em>x<\/em> is representing the number of years since the start of the data\u2014which was 2011, based on the table:<\/p>\n<table>\n<tbody>\n<tr>\n<td>Year<\/td>\n<td>Number of \u00a0High School Students Smoking\u00a0Cigarettes (per 100)<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>16<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>So we are predicting that there will be no smokers in high school by [latex]2011+9.14=2020[\/latex]. How accurate do you think this model is? Do you think there will ever be 0 smokers in high school?<\/p>\n<\/div>\n<p>The following video gives a thorough explanation of making a prediction given a linear equation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-9\" title=\"Make a Prediction Using a Linear Equation (Horizontal Intercept) - Smokers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/5W0qq8saxO0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Bringing it Together<\/h2>\n<p>The last example we will show will include many\u00a0of the concepts that we have been building up throughout this section. \u00a0We will interpret a word problem, write a linear equation from it, graph the equation, interpret the y-intercept and make a prediction. Hopefully this example will help you to make\u00a0connections between the concepts we have presented.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>It costs $600 to purchase an iphone, plus $55 per month for unlimited use and data.<\/p>\n<p>Write a linear equation that represents the cost, y, \u00a0of owning and using the\u00a0iPhone for x amount of months. When you have written your equation, answer the following questions:<\/p>\n<ol>\n<li>What is the total cost you\u2019ve paid after\u00a0owning and using your phone for 24 months?<\/li>\n<li>If you have spent\u00a0$2,580 since you purchased your phone, how many months have you used your phone?<\/li>\n<\/ol>\n<div id=\"attachment_4649\" style=\"width: 216px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4649\" class=\"wp-image-4649\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/07175121\/Screen-Shot-2016-06-07-at-10.50.43-AM-300x220.png\" alt=\"5 iPhones laying next to each other\" width=\"206\" height=\"151\" \/><\/p>\n<p id=\"caption-attachment-4649\" class=\"wp-caption-text\">iPhone<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q282349\">Show Solution<\/span><\/p>\n<div id=\"q282349\" class=\"hidden-answer\" style=\"display: none\"><\/div>\n<\/div>\n<p><strong>Read and Understand:<\/strong>\u00a0We need to write a linear equation that represents the cost of owning and using an iPhone for any number of months. \u00a0We are to use y to represent cost, and x to represent the number of months we have used the phone.<\/p>\n<p><strong>Define and Translate:\u00a0<\/strong>We will use the slope-intercept form of a line, [latex]y=mx+b[\/latex], because we are given a starting cost and a monthly cost for use. \u00a0We will need to find the slope and the y-intercept.<\/p>\n<p>Slope: in this case we don&#8217;t know two points, but we are given a rate in dollars for monthly use of the phone. \u00a0Our units are dollars per month because slope is [latex]\\frac{\\Delta{y}}{\\Delta{x}}[\/latex], and y is in dollars and x is in months. The slope will be [latex]\\frac{55\\text{ dollars }}{1\\text{ month }}[\/latex]. [latex]m=\\frac{55}{1}=55[\/latex]<\/p>\n<p>Y-Intercept: the y-intercept is defined as a point [latex]\\left(0,b\\right)[\/latex]. \u00a0We want to know how much money we have spent, y, after 0 months. \u00a0We haven&#8217;t paid for service yet, but we have paid $600 for the phone. The y-intercept in this case is called an initial cost. [latex]b=600[\/latex]<\/p>\n<p><strong>Write and Solve:\u00a0<\/strong>Substitute the slope and intercept you defined into the slope=intercept equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}y=mx+b\\\\y=55x+600\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Now we will answer the following questions:<\/p>\n<ol>\n<li>What is the total cost you\u2019ve paid after\u00a0owning and using your phone for 24 months?<\/li>\n<\/ol>\n<p>Since x represents the number of months you have used the phone, we can substitute x=24 into our equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}y=55x+600\\\\y=55\\left(24\\right)+600\\\\y=1320+600\\\\y=1920\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Y represents the cost after x number of months, so in this scenario, after 24 months, you have spent $1920 to own and use an iPhone.<\/p>\n<ol>\n<li>If you have spent\u00a0$2,580 since you purchased your phone, how many months have you used your phone?<\/li>\n<\/ol>\n<p>We know that y represents cost, and we are given a cost and asked to find the number of months related to having spent that much. We will substitute y=$2,580 into the equation, then use what we know about solving linear equations to isolate x:<\/p>\n<p style=\"text-align: center;\">\u00a0[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y=55x+600\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2580=55x+600\\\\\\text{ subtract 600 from each side}\\,\\,\\,\\,\\,\\,\\,\\underline{-600}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-600}\\\\\\text{}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,1980=55x\\\\\\text{}\\\\\\text{ divide each side by 55 }\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{1980}{55}=\\frac{55x}{55}\\\\\\text{}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,36=x\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">If you have spent $2,580 then you have been using your iPhone for 36 months, or 3 years.<\/p>\n<\/div>\n<h3>Summary<\/h3>\n<p>The slope-intercept form of a linear equation is written as [latex]y=mx+b[\/latex], where <i>m<\/i> is the slope and <i>b<\/i> is the value of <i>y<\/i> at the <i>y<\/i>-intercept, which can be written as [latex](0,b)[\/latex]. When you know the slope and the <i>y<\/i>-intercept of a line you can use the slope-intercept form to immediately write the equation of that line. The slope-intercept form can also help you to write the equation of a line when you know the slope and a point on the line or when you know two points on the line.<\/p>\n<p>When lines in a plane are parallel (that is, they never cross), they have the same slope. When lines are perpendicular (that is, they cross at a 90\u00b0 angle), their slopes are opposite reciprocals of each other. The product of their slopes will be [latex]-1[\/latex], except in the case where one of the lines is vertical causing its slope to be undefined. You can use these relationships to find an equation of a line that goes through a particular point and is parallel or perpendicular to another line.<\/p>\n<h2><\/h2>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-2686\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Slope-Intercept Form of a Line. <strong>Authored by<\/strong>: Mathispower4u. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><li>Ex: Determine a Linear Equation Given Slope and a Point (Slope-Intercept Form) . <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/URYnKqEctgc\">https:\/\/youtu.be\/URYnKqEctgc<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 1: Find the Equation of a Line in Slope Intercept Form Given Two Points. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/P1ex_a6iYDo\">https:\/\/youtu.be\/P1ex_a6iYDo<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Unit 13: Graphing, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Determine the Equation of a Line Parallel to a Line in General Form. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/TQKz2XHI09E\">https:\/\/youtu.be\/TQKz2XHI09E<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Determine the Equation of a Line Perpendicular to a Line in Slope-Intercept Form. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/QtvtzKjtowA\">https:\/\/youtu.be\/QtvtzKjtowA<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Find the Equation of a Perpendicular and Horizontal Line to a Horizontal Line. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Qpn3f3wMeIs\">https:\/\/youtu.be\/Qpn3f3wMeIs<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Beginning and Intermediate Algebra. <strong>Authored by<\/strong>: Tyler Wallace. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/wallace.ccfaculty.org\/book\/book.html\">http:\/\/wallace.ccfaculty.org\/book\/book.html<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":20,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Beginning and Intermediate Algebra\",\"author\":\"Tyler Wallace\",\"organization\":\"\",\"url\":\" http:\/\/wallace.ccfaculty.org\/book\/book.html\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Slope-Intercept Form of a Line\",\"author\":\"Mathispower4u\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"pd\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Determine a Linear Equation Given Slope and a Point (Slope-Intercept Form) \",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/URYnKqEctgc\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 1: Find the Equation of a Line in Slope Intercept Form Given Two Points\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/P1ex_a6iYDo\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 13: Graphing, from Developmental Math: An Open 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