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<h4>Activity (20 minutes)</h4>
<p>Previously in this unit, students have practiced solving for a particular variable and also connected equations to graphs. In earlier lessons, they worked to write the equation of a line from tables, graphs, and given information. Students have already written equations in slope-intercept form given a point and slope and in point-slope form given two points. Now, they will work to write linear equations in slope-intercept form given two points using point-slope form and not using point-slope form in the process.</p>
<p>They will need to activate prior knowledge of using the slope formula as well as the forms used and the meaning of the variables in each form.</p>
<h4>Launch</h4>
<p>Divide students into pairs. Have students work independently on questions 1-3 then compare and discuss answers and work. Then have them repeat for 4-7. For questions 8-12, have students work in their pairs.</p>
<br>
<h4>Student Activity</h4>
<br>
<p><strong>Writing Equations Given Two Points</strong></p>
<p>For questions 1-3, use the point \((3,4)\) and \((-2, 5)\).</p>
<ol class="os-raise-noindent" >
<li> Find the slope of the line that goes through the two points. </li>
</ol>
<p><strong>Answer:</strong> \(-\frac15\)<br>
\(m=\frac{5-4}{-2-3}=\frac{1}{-5}\)</p>
<ol class="os-raise-noindent" start="2">
<li> Find the equation in point slope-form of the line that goes through the two points. </li>
</ol>
<p><strong>Answer:</strong> \(y-y_1=m(x-x_1)\)<br>
\(y-4=-\frac{1}{5}(x-3)\)</p>
<ol class="os-raise-noindent" start="3">
<li> Write the equation from question 2 into slope-intercept form. </li>
</ol>
<p>(Hint: Solve for \(y\).)</p>
<p><strong>Answer:</strong> <br>
\(\begin{array}{rcl}y&=&-\frac{1}{5}x+\frac{23}{5}\\y-4&=&-\frac{1}{5}x+\frac35\\y-4+4&=&-\frac{1}{5}x+\frac{3}{5}+4\\y-4+4&=&-\frac{1}{5}x+\frac{3}{5}+\frac{20}{5}\\y&=&-\frac{1}{5}x+\frac{23}{5}\end{array}\)</p>
<br>
<p><strong>Writing Equations in Slope-Intercept Form Given a Point and Slope</strong></p>
<p>For questions 4-7 use the point \((3, -2)\) and the slope -4.</p>
<ol class="os-raise-noindent" start="4">
<li> Substitute the slope into slope-intercept form. </li>
</ol>
<p><strong>Answer:</strong> \(y= -4x +b\)</p>
<ol class="os-raise-noindent" start="5">
<li> Substitute the given point in for \((x, y)\) in the equation. </li>
</ol>
<p><strong>Answer:</strong> \(-2= -4(3) +b\)</p>
<ol class="os-raise-noindent" start="6">
<li> Solve for \(b\) in slope-intercept form. </li>
</ol>
<p><strong>Answer:</strong> 10</p>
<ol class="os-raise-noindent" start="7">
<li> Write the equation in slope-intercept form. </li>
</ol>
<p><strong>Answer:</strong> \(y= -4x +10\)</p>
<br>
<p>With a partner, work to go directly from two points to an equation in slope-intercept form, without using point-slope form.</p>
<p>For questions 8-12, use the two points \((5, 2)\) and \((3, -1)\)</p>
<ol class="os-raise-noindent" start="8">
<li> Find the slope between the two points. </li>
</ol>
<p><strong>Answer:</strong> \(\frac32\)<br>
\(m=\frac{-1-2}{3-5}=\frac{-3}{-2}=\frac32\)</p>
<ol class="os-raise-noindent" start="9">
<li> Substitute the slope into slope-intercept form. </li>
</ol>
<p><strong>Answer:</strong> \(y= \frac{3}{2}x +b\)</p>
<ol class="os-raise-noindent" start="10">
<li> Substitute the one point in for \((x, y)\) in the equation. </li>
</ol>
<p><strong>Answer:</strong> \(5= \frac{3}{2}(2) +b\)</p>
<ol class="os-raise-noindent" start="11">
<li> Solve for \(b\) in slope-intercept form. </li>
</ol>
<p><strong>Answer:</strong> 2</p>
<ol class="os-raise-noindent" start="12">
<li> Write the equation in slope-intercept form. </li>
</ol>
<p><strong>Answer:</strong> \(y= \frac{3}{2}x +2\)</p>
<br>
<h4>Activity Synthesis</h4>
<p>Questions 1-3 reinforce the work that was done in the warm up.</p>
<p>For question 10, ask if any students used the other point for substitution. Show students the comparison of the work when each point is used. Emphasize that since both points are on the line, either point works. </p>
<p>Ask: “What was the advantage of using \((5, 2)\) for this problem?” (The fraction in the slope canceled making it easier to solve for \(b\).)</p>
<p>If time allows, have students connect back to graphing by graphing the lines in the activity. Compare how negative versus positive slopes change the graphs. Have students identify the \(y\)-intercept of each line.</p>
<br>
<br>
<h4>1.14.2: Self Check</h4>
<p class="os-raise-text-bold"><em>After the activity, students will answer the following question to check their understanding of the concepts explored in the activity.</em></p>
<p class="os-raise-text-bold">QUESTION:</p>
<p>Which of the following is the equation of a line that has a slope of 3 and goes through the point \((1, -2)\)?</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>\(y= 3x -2 \)</td>
<td>Incorrect. Let’s try again a different way: Substitute the slope in for \(m\) and the point \((1, -2)\) for \((x, y)\) in \(y=mx +b\). This becomes \(-2=3(1) +b\). The answer is:\( y = 3x - 5\). </td>
</tr>
<tr>
<td>\( y = 3x +1\)</td>
<td> Incorrect. Let’s try again a different way: When solving for \(b\), make sure to use inverse operations. The answer is: \(y = 3x - 5\).</td>
</tr>
<tr>
<td>\(y = 3x - 5\)</td>
<td> That’s correct! Check yourself: Substitute the slope in for \(m\) and the point \((1, -2)\) for \((x, y)\) in \(y=mx +b\). This becomes \(-2=3(1) +b\). Then solve for \(b\) which is \(-5\). </td>
</tr>
<tr>
<td>\(y= 1x -2\)</td>
<td> Incorrect. Let’s try again a different way: Substitute the slope in for \(m\) and the point \((1, -2)\) for \((x, y)\) in \(y=mx +b\). This becomes \(-2=3(1) +b\). The answer is: \(y = 3x - 5\). </td>
</tr>
</tbody>
</table>
<br>
<br>
<h4>1.13.2: Additional Resources </h4>
<p><strong><em>The following content is available to students who would like more support based on their experience with the self check. Students will not automatically have access to this content, so you may wish to share it with those who could benefit from it. </em></strong></p>
<br>
<h4>Find an Equation Given Two Points</h4>
<h5>Method 1: Using Point-Slope Form</h5>
<p>Find an equation of a line that contains the points \((5, 4)\) and \((3, 6)\). Write the equation in slope-intercept form.</p>
<p><strong>Step 1</strong> - Find the slope between the two points.<br>
\(m=\frac{6-4}{3-5}=\frac{2}{-2}=-1\)</p>
<p><strong>Step 2</strong> - Choose a point to substitute into point-slope form.<br>
\(y-y_1=m(x-x_1)\)<br>
\(y-4=-1(x-5)\) OR \(y-6=-1(x-3) \)</p>
<p><strong>Step 3</strong> - Write the equation in slope-intercept form.</p>
<div class="os-raise-d-flex os-raise-justify-content-evenly">
<p>\(\begin{array}{rcl}y-4&=&-1(x-5)\\y-4&=&-1x+5\\y&=&-x +9\end{array}\)</p>
<p>OR</p>
<p>\(\begin{array}{rcl}y-6&=&-1(x-3)\\ y-6&=&-1x+3\\y&=&-x +9\end{array}\)</p>
</div>
<br>
<p><strong>Method 2: Writing an Equation in Slope-Intercept Form by finding \(b\).</strong></p>
<p>Find the equation of the line containing the points \((3, 1)\) and \((5, 6)\) in slope-intercept form.</p>
<p><strong>Step 1</strong> - Find the slope between the two points.<br>
\(m=\frac{6-1}{5-3}=\frac{5}{2}\)</p>
<p><strong>Step 2</strong> - Choose a point to substitute into slope-intercept form then solve for \(b\). <br>
\(y=mx+b\)</p>
<div class="os-raise-d-flex os-raise-justify-content-evenly">
<p>\(\begin{array}{rcl}y&=&\frac{5}{2}x+b\\1&=&\frac{5}{2}(3)+b\\\frac{2}{2}&=&\frac{15}{2}+b\\b&=&\frac{13}{2}\end{array}\)</p>
<p> OR</p>
<p> \(\begin{array}{rcl}y&=&\frac{5}{2}x+b\\6&=&\frac{5}{2}(5)+b\\\frac{12}{2}&=&\frac{25}{2}+b\\b&=&\frac{13}{2}\end{array}\)</p>
</div>
<br>
<p><strong>Step 3</strong> - Write the equation in slope-intercept form.<br>
\(y=\frac{5}{2}x+\frac{13}{2}\)</p>
<br>
<h4>Try It: Find an Equation Given Two Points</h4>
<p>Find the equation of a line that consists of the points \((1, 4)\) and \((6, 2)\) in slope-intercept form.</p>
<p><strong>Answer:</strong> \(y=-\frac{2}{5}x+\frac{22}{5}\)</p>
<p><strong>Step 1</strong> - Find the slope between the two points.<br>
\(m=\frac{2-4}{6-1}=\frac{-2}{5}\)</p>
<p><strong>Step 2</strong> - Choose a point to substitute into slope-intercept form then solve for \(b\). <br>
\(y=mx+b\)</p>
<div class="os-raise-d-flex os-raise-justify-content-evenly">
<p>\(\begin{array}{rcl}y&=&-\frac{2}{5}x+b\\4&=&-\frac{2}{5}(1)+b\\\frac{20}{5}&=&-\frac{2}{5}+b\\b&=&\frac{22}{5}\end{array}\)</p>
<p>OR</p>
<p>\(\begin{array}{rcl}y&=&-\frac{2}{5}x+b\\2&=&-\frac{2}{5}(6)+b\\\frac{10}{5}&=&-\frac{12}{5}+b\\b&=&\frac{22}{5}\end{array}\)</p>
</div>
<br>
<p><strong>Step 3</strong> - Write the equation in slope-intercept form.<br>
\(y=-\frac{2}{5}x+\frac{22}{5}\) </p>