<h2>Using $m=\frac{\text{rise}}{\text{run}}$ to Find the Slope of a Line From Its Graph</h2>
<p>Now, we’ll look at some graphs on the $xy$-coordinate plane and see how to find their slopes. The method will be very similar to what we just modeled on our geoboards.</p>
<p>To find the slope, we must count out the rise and the run. But where do we start?</p>
<p>We locate two points on the line whose coordinates are integers. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.</p>
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<h3>Example: How to Use $m=\frac{\text{rise}}{\text{run}}$ to Find the Slope of a Line from its Graph</h3>
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<p>Find the slope of the line shown.</p>
<img alt="The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 6 and the y-axis runs from negative 4 to 2. A line passes through the points (0, negative 3) and (5, 1)." src="https://data.ulearngo.com/assets/8a0f6f25-84cb-4048-a4ca-fd6b353d67ca"/>
<h3 data-type="title">Solution</h3>
<div data-type="title"><img alt="This table has three columns and four rows. The first row says, “Step 1. Locate two points on the graph whose coordinates are integers. Mark (0, negative 3) and (5, 1).” To the right is a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 6. The y-axis of the plane runs from negative 4 to 2. The points (0, negative 3) and  (5, 1) are plotted." src="https://data.ulearngo.com/assets/0880fd27-95b4-4072-b60e-da8e725b8020"/><img alt="The second row says, “Step 2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point. Starting at (0, negative 3), sketch a right triangle to (5, 1).” In the graph on the right, an additional point is plotted at (0, 1). The three points form a right triangle, with the line from (0, negative 3) to (5, 1) forming the hypotenuse and the lines from (0, negative 3) to (0, 1) and (0, 1) to (5, 1) forming the legs." src="https://data.ulearngo.com/assets/d75e831e-28b7-4f0d-9aa2-3ba0641e36f0"/><img alt="The third row then says, “Step 3. Count the rise and the run on the legs of the triangle.” The rise is 4 and the run is 5." src="https://data.ulearngo.com/assets/9f953c23-0d3d-4832-9c49-b3cc755242c9"/><img alt="The fourth row says, “Step 4. Take the ratio of the rise to run to find the slope. Use the slope formula. Substitute the values of the rise and run.” To the right is the slope formula, m equals rise divided by run. The slope of the line is 4 divided by 5, or four fifths. This means that y increases 4 units as x increases 5 units." src="https://data.ulearngo.com/assets/800e8d0e-c563-4fdc-a1af-3f877cb7937c"/></div>
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<div class="ns-school-defn2">
<header>
<h3 data-type="title">Find the slope of a line from its graph using $m=\frac{\text{rise}}{\text{run}}.$</h3>
</header>
<ol type="1">
<li>Locate two points on the line whose coordinates are integers.</li>
<li>Starting with the point on the left, sketch a right triangle, going from the first point to the second point.</li>
<li>Count the rise and the run on the legs of the triangle.</li>
<li>Take the ratio of rise to run to find the slope, $m=\frac{\text{rise}}{\text{run}}$.</li>
</ol>
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<h2>Example</h2>
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<div data-type="example">
<div>
<div>
<p>Find the slope of the line shown.</p>
<img alt="The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 9 and the y-axis runs from negative 1 to 7. A line passes through the points (0, 5), (3, 3), and (6, 1)." src="https://data.ulearngo.com/assets/9a36659f-4104-44f7-a9d7-0682de97e10b"/></div>
<div>
<h3>Solution</h3>
<table summary="This figure shows step-by-step how to find the slope of the line with points (0, 5) and (3, 3). First, identify the leftmost point, which is (0, 5). Starting at (0, 5), sketch a right triangle to (3, 3). To the right is the line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 9. The y-axis of the plane runs from negative 1 to 7. The points (0, 5) and  (3, 3) are plotted. An additional point is plotted at (0, 3). The three points form a right triangle, with the line from (0, 5) to (3, 3) forming the hypotenuse and the lines from (0, 5) to (0, 3) and from (0, 3) to (3, 3) forming the legs. The leg from (0, 5) to (0, 3) is labeled “rise” and the leg from (0, 3) to (3, 3) is labeled “run”. The next step is to count the rise, which is negative. The rise is negative 2. The next step is to count the run, which is 3. Now use the slope formula, m equals rise over run. Substitute the values of the rise and run to get m equals negative 2 thirds.">
<tbody>
<tr>
<td data-valign="top">Locate two points on the graph whose coordinates are integers.</td>
<td data-valign="top">$\left(0,5\right)$ and $\left(3,3\right)$</td>
</tr>
<tr>
<td data-valign="top">Which point is on the left?</td>
<td data-valign="top">$\left(0,5\right)$</td>
</tr>
<tr>
<td data-valign="top">Starting at $\left(0,5\right)$, sketch a right triangle to $\left(3,3\right)$.</td>
<td data-valign="top"><img alt="." src="https://data.ulearngo.com/assets/465e90db-365e-477b-b9e9-b581cbe452ad"/></td>
</tr>
<tr>
<td data-valign="top">Count the rise—it is negative.</td>
<td data-valign="top">The rise is $-2$.</td>
</tr>
<tr>
<td data-valign="top">Count the run.</td>
<td data-valign="top">The run is 3.</td>
</tr>
<tr>
<td data-valign="top">Use the slope formula.</td>
<td data-valign="top">$m=\frac{\text{rise}}{\text{run}}$</td>
</tr>
<tr>
<td data-valign="top">Substitute the values of the rise and run.</td>
<td data-valign="top">$m=\frac{-2}{3}$</td>
</tr>
<tr>
<td data-valign="top">Simplify.</td>
<td data-valign="top">$m=-\frac{2}{3}$</td>
</tr>
<tr>
<td data-valign="top"> </td>
<td data-valign="top">The slope of the line is $-\frac{2}{3}$.</td>
</tr>
</tbody>
</table>
<p>So $y$ increases by 3 units as $x$ decreases by 2 units.</p>
<p>What if we used the points $\left(-3,7\right)$ and $\left(6,1\right)$ to find the slope of the line?</p>
<img alt="The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 3, 7) and (6, 1). An additional point is plotted at (negative 3, 1). The three points form a right triangle, with the line from (negative 3, 7) to (6, 1) forming the hypotenuse and the lines from (negative 3, 7) to negative 1, 7) and from (negative 1, 7) to (6, 1) forming the legs." src="https://data.ulearngo.com/assets/d4493247-ea2a-4373-81ae-eef3d4d6d39d"/>
<p>The rise would be $-6$ and the run would be 9. Then $m=\frac{-6}{9}$, and that simplifies to $m=-\frac{2}{3}$. Remember, it does not matter which points you use—the slope of the line is always the same.</p>
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<p>In the last two examples, the lines had <em>y</em>-intercepts with integer values, so it was convenient to use the <em>y</em>-intercept as one of the points to find the slope. In the next example, the <em>y</em>-intercept is a fraction. Instead of using that point, we’ll look for two other points whose coordinates are integers. This will make the slope calculations easier.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Find the slope of the line shown.</p>
<img alt="The graph shows the x y coordinate plane. The x-axis runs from 0 to 8 and the y-axis runs from 0 to 7. A line passes through the points (2, 3) and (7, 6)." src="https://data.ulearngo.com/assets/a83f4335-6aa8-4308-9ba4-a35af019cde4"/></div>
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<h3>Solution</h3>
<table summary="This figure shows step-by-step how to find the slope of the line with points (2, 3) and (7, 6). First, identify the leftmost point, which is (2, 3). Starting at (2, 3), sketch a right triangle to (7, 6). To the right is the line graphed on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis of the plane runs from 0 to 7. The points (2, 3) and  (7, 6) are plotted. An additional point is plotted at (2, 6). The three points form a right triangle, with the line from (2, 3) to (7, 6) forming the hypotenuse and the lines from (2, 3) to (2, 6) and from (2, 6) to (7, 6) forming the legs. The leg from (2, 3) to (2, 6) is labeled “rise” and the leg from (2, 6) to (7, 6) is labeled “run”. The next step is to count the rise, which is 3. The next step is to count the run, which is 5. Now use the slope formula, m equals rise over run. Substitute the values of the rise and run to get m equals 3 fifths.">
<tbody>
<tr>
<td data-valign="top">Locate two points on the graph whose coordinates are integers.</td>
<td data-valign="top">$\left(2,3\right)$ and $\left(7,6\right)$</td>
</tr>
<tr>
<td data-valign="top">Which point is on the left?</td>
<td data-valign="top">$\left(2,3\right)$</td>
</tr>
<tr>
<td data-valign="top">Starting at $\left(2,3\right)$, sketch a right triangle to $\left(7,6\right)$.</td>
<td data-valign="top"><img alt="." src="https://data.ulearngo.com/assets/3ed328c1-d51a-4e0c-9350-696d1bcfb9b1"/></td>
</tr>
<tr>
<td data-valign="top">Count the rise.</td>
<td data-valign="top">The rise is 3.</td>
</tr>
<tr>
<td data-valign="top">Count the run.</td>
<td data-valign="top">The run is 5.</td>
</tr>
<tr>
<td data-valign="top">Use the slope formula.</td>
<td data-valign="top">$m=\frac{\text{rise}}{\text{run}}$</td>
</tr>
<tr>
<td data-valign="top">Substitute the values of the rise and run.</td>
<td data-valign="top">$m=\frac{3}{5}$</td>
</tr>
<tr>
<td data-valign="top"> </td>
<td data-valign="top">The slope of the line is $\frac{3}{5}$.</td>
</tr>
</tbody>
</table>
<p>This means that $y$ increases 5 units as $x$ increases 3 units.</p>
<p>When we used geoboards to introduce the concept of slope, we said that we would always start with the point on the left and count the rise and the run to get to the point on the right. That way the run was always positive and the rise determined whether the slope was positive or negative.</p>
<p>What would happen if we started with the point on the right?</p>
<p>Let’s use the points $\left(2,3\right)$ and $\left(7,6\right)$ again, but now we’ll start at $\left(7,6\right)$.</p>
<img alt="The graph shows the x y coordinate plane. The x -axis runs from 0 to 8. The y -axis runs from 0 to 7. A line passes through the points (2, 3) and (7, 6). An additional point is plotted at (7, 3). The three points form a right triangle, with the line from (2, 3) to (7, 6) forming the hypotenuse and the lines from (2, 3) to (7, 3) and from (7, 3) to (7, 6) forming the legs." src="https://data.ulearngo.com/assets/6b27940b-b568-4142-a898-c90ebedbb8ee"/>
<p>$\begin{array}{ccc}\text{Count the rise.}\hfill &amp; &amp; \text{The rise is}\phantom{\rule{0.2em}{0ex}}-3.\hfill \\ \begin{array}{c}\text{Count the run. It goes from right to left, so}\hfill \\ \text{it is negative.}\hfill \end{array}\hfill &amp; &amp; \text{The run is}\phantom{\rule{0.2em}{0ex}}-5.\hfill \\ \text{Use the slope formula.}\hfill &amp; &amp; m=\frac{\text{rise}}{\text{run}}\hfill \\ \text{Substitute the values of the rise and run.}\hfill &amp; &amp; m=\frac{-3}{-5}\hfill \\ &amp; &amp; \text{The slope of the line is}\phantom{\rule{0.2em}{0ex}}\frac{3}{5}.\hfill \end{array}$</p>
<p>It does not matter where you start—the slope of the line is always the same.</p>
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Using $m=\frac{\text{rise}}{\text{run}}$ to Find the Slope of a Line From Its Graph

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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.


Using $m=\frac{\text{rise}}{\text{run}}$ to Find the Slope of a Line From Its Graph

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