<h2>Finding the Equation of a Line</h2>
<p>How do online retailers know that ‘you may also like’ a particular item based on something you just ordered? How can economists know how a rise in the minimum wage will affect the unemployment rate? How do medical researchers create drugs to target cancer cells? How can traffic engineers predict the effect on your commuting time of an increase or decrease in gas prices? It’s all mathematics.</p>
<p>You are at an exciting point in your mathematical journey as the mathematics you are studying has interesting applications in the real world.</p>
<p>The physical sciences, social sciences, and the business world are full of situations that can be modeled with linear equations relating two variables. Data is collected and graphed. If the data points appear to form a straight line, an equation of that line can be used to predict the value of one variable based on the value of the other variable.</p>
<p>To create a mathematical model of a linear relation between two variables, we must be able to find the equation of the line. In this lesson and the next few, we will look at several ways to write the equation of a line. The specific method we use will be determined by what information we are given.</p>
<h2>Finding an Equation of the Line Given the Slope and <em>y</em>-Intercept</h2>
<p>We can easily determine the slope and intercept of a line if the equation was written in slope–intercept form, \(y=mx+b.\) Now, we will do the reverse—we will start with the slope and <em>y</em>-intercept and use them to find the equation of the line.</p>
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<h2>Example</h2>
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<p>Find an equation of a line with slope \(-7\) and <em>y</em>-intercept \(\left(0,-1\right)\).</p>
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<h3>Solution</h3>
<p>Since we are given the slope and <em>y</em>-intercept of the line, we can substitute the needed values into the slope–intercept form, \(y=mx+b\).</p>
<table summary="This figure has two columns. In the top row, the instructions in the left column say “Name the slope.” In the right column is m equals negative 7. One row down, the instructions in the left column say “Name the y-intercept.” In the right column is the y-intercept defined as the ordered pair (0, negative 1). One row down, the instructions in the left column say “Substitute the values into y equals mx plus b.” In the right column is the slope-intercept form: y equals mx plus b. Below this is the formula with negative 7 substituted for m and negative 1 substituted for b: y equals negative 7x plus negative 1. Below this is the formula simplified: y equals negative 7x minus 1.">
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<td style="width: 317px;">Name the slope.</td>
<td style="width: 166px;"><img alt="." src="https://data.ulearngo.com/assets/b65fc125-7bde-4e78-a6c3-98d4736a7748"/></td>
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<td style="width: 317px;">Name the <em>y</em>-intercept.</td>
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<td style="width: 317px;">Substitute the values into \(y=mx+b.\)</td>
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<td style="width: 317px;"> </td>
<td style="width: 166px;"><img alt="." src="https://data.ulearngo.com/assets/35b3e679-a115-4851-8ea4-bfb589c045dd"/></td>
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<td style="width: 317px;"> </td>
<td style="width: 166px;"><img alt="." src="https://data.ulearngo.com/assets/8f7419b6-b7b4-4329-9006-dccf85ab2538"/></td>
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<p>Sometimes, the slope and intercept need to be determined from the graph.</p>
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<h2>Example</h2>
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<p>Find the equation of the line shown.</p>
<img alt="The graph shows the x y-coordinate plane. The x and y-axes each run from negative 7 to 7. A line intercepts the y-axis at (0, negative 4), passes through the plotted point (3, negative 2), and intercepts the x-axis at (4, 0)." src="https://data.ulearngo.com/assets/94b5fba4-d513-4375-abba-b9e16b7a17ea"/></div>
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<h3>Solution</h3>
<p>We need to find the slope and <em>y</em>-intercept of the line from the graph so we can substitute the needed values into the slope–intercept form, \(y=mx+b\).</p>
<p>To find the slope, we choose two points on the graph.</p>
<p>The <em>y</em>-intercept is \(\left(0,-4\right)\) and the graph passes through \(\left(3,-2\right)\).</p>
<table summary="This figure has two columns. In the top row, the instructions in the left column say “Find the slope by counting the rise and run.” In the right column is m equals rise over run. Under that is m equals two thirds. One row down, the instructions in the left column say “Find the y-intercept.” In the right column is the y-intercept defined as the ordered pair (0, negative 4). One row down, the instructions in the left column say “Substitute the values into y equals mx plus b.” In the right column is the slope-intercept form: y equals mx plus b. Below this is the formula with negative two thirds substituted for m and negative 4 substituted for b: y equals negative two thirds x minus 4.">
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<td style="width: 324px;">Find the slope by counting the rise and run.</td>
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<td style="width: 324px;"> </td>
<td style="width: 165px;"><img alt="." src="https://data.ulearngo.com/assets/d06f70fe-607d-467b-8ce1-7e476b6a2da3"/></td>
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<td style="width: 324px;">Find the <em>y</em>-intercept.</td>
<td style="width: 165px;"><img alt="." src="https://data.ulearngo.com/assets/502aca9a-e74e-4d47-949b-9bcb63200247"/></td>
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<td style="width: 324px;">Substitute the values into \(y=mx+b.\)</td>
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<td style="width: 324px;"> </td>
<td style="width: 165px;"><img alt="." src="https://data.ulearngo.com/assets/1a3e7869-5bab-4289-8b7a-ad31ada034bc"/></td>
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Finding an Equation of the Line Given the Slope and <em>y</em>-intercept

Mathematics studies measurement, relationships, and properties of quantities and sets, using numbers and symbols. Arithmetic, algebra, geometry, and calculus are popular topics in mathematics.

Mathematics

Explore how to graph linear equations, find solutions to equations and inequalities, identify intercepts on a graph, model slope using geoboards, and find equations of lines using slope and intercepts or given points.


Finding an Equation of the Line Given the Slope and <em>y</em>-intercept

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