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<h4>Activity (10 minutes)</h4>
<p>In the warm up, students saw that some systems have infinitely many solutions. In this activity, they encounter a situation that can be represented with a system of equations, but the system has no solutions. Students write equations to represent the two constraints in the situation and then solve the system algebraically and graphically.</p>
<p>As students work, notice the different ways they reach the conclusion that the systems have no solutions. Identify students with varying strategies and ask them to share later.</p>
<h4>Launch</h4>
<p>Keep students in groups of 3–4 and provide access to graphing technology.</p>
<h4>Student Activity</h4>
<p>A recreation center is offering special prices on its pool passes and gym memberships for the summer. On the first day of the offering, a family paid $96 for 4 pool passes and 2 gym memberships. Later that day, an individual bought a pool pass for herself, a pool pass for a friend, and 1 gym membership. She paid $72.</p>
<ol class="os-raise-noindent">
<li> Write a system of equations that represents the relationships between pool passes, gym memberships, and the costs. Be sure to state what each variable represents. </li>
</ol>
<p><strong>Answer:</strong></p>
<p>\(\left\{\begin{array}{rcl}4p+2g&=&96 \\ 2p+1g&=&72\end{array}\right.\)</p>
<p>where \(p\) is the price of each pool pass and \(g\) is the price of a gym membership for 1 person.</p>
<ol class="os-raise-noindent" start="2">
<li> Find the price of a pool pass and the price of a gym membership by solving the system algebraically. Be prepared to show your reasoning. </li>
</ol>
<p><strong>Answer:</strong> The system has no solutions. For example: Solving by elimination or substitution leads to a false equation (such as \(0=24\)).</p>
<ol class="os-raise-noindent" start="3">
<li> Use the graphing tool or technology outside the course. Graph the equations in the system using the Desmos tool below. </li>
</ol>
<p> Students were provided access to Desmos. </p>
<p><strong>Answer:</strong>
<p><img alt="Graph of a linear system. price for a gym membership, price for a pool pass." src="https://k12.openstax.org/contents/raise/resources/dd3206b7b6fb91c32117842c5dd999b9ea5fe0e8"></p>
<ol class="os-raise-noindent" start="4">
<li>Then make 1–2 observations about your graphs.</li>
</ol>
<p><strong>Answer:</strong> Your answer may vary, but here is a sample.</p>
<ul>
<li>The graphs are two parallel lines that don’t intersect. </li>
<li> The lines have the same slope but different vertical intercepts. </li>
</ul>
<ol class="os-raise-noindent" start="5">
<li>How do your observations about the graph of this system relate to the fact that the system was determined to not have a solution?</li>
</ol>
<p><strong>Answer:</strong> Answers will vary. </p>
<br>
<h4>Video: Analyzing Systems of Linear Equations with No
Solution</h4>
<p>Watch the following
video to learn more about systems that have no solution.</p>
<div class="os-raise-d-flex-nowrap os-raise-justify-content-center">
<div class="os-raise-video-container"><video controls="true" crossorigin="anonymous">
<source src="https://k12.openstax.org/contents/raise/resources/0ea17785bff47e386d2c972d61fc943876896f4f">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/3493dc4853b107ba20b395227ef7a18b916aaf06" srclang="en_us">
https://k12.openstax.org/contents/raise/resources/0ea17785bff47e386d2c972d61fc943876896f4f
</video></div>
</div>
<br>
<br>
<br>
<h3>Activity Synthesis</h3>
<p>Invite previously identified students to share their response to the second question. Record or display their reasoning for all to see. After each student shares, ask if anyone else reasoned the same way.<br>
</p>
<p>Next, select other students to share their observations about the graphs. Ask students:</p>
<ul>
<li>“How can we tell from the graphs that there are no solutions?” (The lines are parallel.)</li>
<li>“How can we tell for sure that the lines are parallel and never intersect?” (The slope of both graphs is \(-2\), but they have different intercepts.)</li>
<li>“Why do parallel lines mean no solutions?” (A solution is a pair of values that satisfy both equations and are on both graphs. There are no points that are on both lines simultaneously.)</li>
<li>“What does ‘no solutions’ mean in this situation, in terms of price of pool passes and gym memberships?” (The prices for a pool pass and for a gym membership are different for the two purchases.)</li>
</ul>
<p>Here are some ways to think about the situation:</p>
<ul>
<li>The family purchased twice the number of of pool passes and gym memberships as the individual did, but they did not pay twice as much, so the prices of passes and memberships must have been different for the two purchases.</li>
<li>The person who bought half as many passes and memberships did not pay half as much, which meant that different prices applied to the two transactions.</li>
<li>The special rates for a family of 4 did not apply to the individual, hence the different prices.</li>
</ul>
<br>
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for English Language Learners</p>
<p class="os-raise-extrasupport-name">MLR 6 Three Reads: Reading, Listening, Conversing</p>
</div>
<div class="os-raise-extrasupport-body">
<p>Use this routine to support reading comprehension of this problem. Ask students to keep their books or devices closed and display only the image and the task statement without revealing the questions that follow. Use the first read to orient students to the situation. After a shared reading, ask students: “What is this situation about?” (A recreation center sells pool passes and gym memberships.) After the second read, students list any quantities that can be counted or measured, without focusing on specific values (total amount of money paid by the family and by the individual, the number of pool passes and number of gym memberships purchased by each). During the third read, the questions are revealed. Invite students to discuss possible strategies, referencing the relevant quantities named after the second read.</p>
<p class="os-raise-text-italicize">Design Principle: Support sense-making</p>
<p class="os-raise-extrasupport-title">Learn more about this routine</p>
<p>
<a href="https://www.youtube.com/watch?v=Q2PGJThrG2Q;&rel=0" target="_blank">View the instructional video</a>
and
<a href="https://k12.openstax.org/contents/raise/resources/bf750b41e6483d334d575e1d950851bfa07cfd26" target="_blank">follow along with the materials</a>
to assist you with learning this routine.
</p>
<p class="os-raise-extrasupport-title">Provide support for students</p>
<p>
<a href="https://k12.openstax.org/contents/raise/resources/06eafa198e5345452a0adbc81731e7383785aa42" target="_blank">Distribute graphic organizers</a>
to the students to assist them with participating in this routine.
</p>
</div>
</div>
<br>
<br>
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for Students with Disabilities</p>
<p class="os-raise-extrasupport-name">Representation: Access for Perception</p>
</div>
<div class="os-raise-extrasupport-body">
<p>
Read the “What’s the Deal?” scenario aloud. Students who both listen to and read the information will benefit from extra processing time. Encourage students to annotate their document as they follow along. </p>
<p class="os-raise-text-italicize">Supports accessibility for: Conceptual processing</p>
</div>
</div>
<br>
<br>
<h4>2.7.2: 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>Solve the system of equations
using elimination.<br>
\( \left\{ \begin{array}{rcl} -0.1x - 0.2y &=& 0.6\\ -5x - 10y &=& 1 \end{array}\right. \)</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>(\(0.5, -0.3)\)</td>
<td>Incorrect. Let’s try again a different way: <br>
Multiply the first equation by \( 50 \) to get \( -5x - 10y = 30 \). <br>
Add the equations together. The result is \( 0 = 31 \). <br>
Since this is not true, the answer is no solution.<br></td>
</tr>
<tr>
<td>(\( \frac{1}{3}, 5 \))</td>
<td>Incorrect. Let’s try again a different way:<br>
Multiply the first equation by \( 50 \) to get \( -5x - 10y = 30 \).<br>
Add the equations together. The result is \( 0 = 31 \).<br>
Since this is not true, the answer is no solution.) <br></td>
</tr>
<tr>
<td>This system of equations has infinite solutions.</td>
<td>Incorrect. Let’s try again a different way:<br>
Multiply the first equation by \( 50 \) to get \( -5x - 10y = 30 \).<br>
Add the equations together. The result is \( 0 = 31 \).<br>
Since this is not true, the answer is no solution.<br></td>
</tr>
<tr>
<td>This system of equations has no solution.</td>
<td>That’s correct! Check yourself: <br>
If you multiply the first equation by 50 and add, the result is \( 0 = 31 \). <br>
Since this
is not true, the answer is no solution.<br></td>
</tr>
</tbody>
</table>
<br>
<h4>2.7.2: Additional Resources</h4>
<p class="os-raise-text-bold"><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></p>
<h4>Solving an Inconsistent System of Equations</h4>
<p>Now that we have several methods for solving systems of equations, we can use the methods to identify inconsistent systems. Recall that an inconsistent system consists of parallel lines that have the same slope but different \(y\)-intercepts. They will never intersect. When searching for a solution to an inconsistent system, we will come up with a false statement, such as \( 12 = 0 \).<br>
</p>
<p>Solve the following system of equations. </p>
<p>\( x = 9 - 2y \) <br>
\( x+2y =13 \)</p>
<p><strong>Solution</strong> </p>
<p>We can approach this problem in two ways. Because one equation is already solved for \( x \), the most obvious step is to use substitution. </p>
<p>\( \begin{array}{rcl}x+2y &=&13 \\ (9-2y)+2y &=&13 \\ 9+0y &=&13 \\ 9&=&13\end{array} \)</p>
<p>Clearly, this statement is a contradiction because \( 9 \neq 13 \). Therefore, the system has no solution. </p>
<p>The second approach would be to first manipulate the equations so that they are both in slope-intercept form. We manipulate the first equation as follows:</p>
<p>\( x = 9 - 2y \)<br>
\( 2y = -x + 9x\)<br>
\( y= -\frac{1}{2}x + \frac{9}{2} \)</p>
<p>We then convert the second equation expressed to slope-intercept form. </p>
<p>\( \begin{array}{rcl}x + 2y &=& 13 \\ 2y &=& -x + 13 \\ y &=& -\frac{1}{2}x + \frac{13}{2}\end{array} \)</p>
<p>Comparing the equations, we see that they have the same slope but different \(y\)-intercepts. Therefore, the lines are parallel and do not intersect. </p>
<p>\( y= -\frac{1}{2}x + \frac{9}{2} \)<br>
\( y= -\frac{1}{2}x + \frac{13}{2} \)</p>
<h4>Try It: Solving an Inconsistent System of Equations</h4>
<p>Solve the following system of equations. If there is no solution, explain how you know.</p>
<p>\( 9x +3y = -24 \)<br>
\( 6x +2y = 14 \)</p>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answer:</strong></p>
<p>Here is how to solve this system of equations:</p>
<p>To solve using elimination, we must first make the coefficients of one of the variables the same to use subtraction.</p>
<p>\( 2(9x +3y) = 2(-24) \)<br>
\( 3(6x +2y) = 3(14) \)</p>
<p>After simplifying, we have:</p>
<p>\( 18x +6y = -48 \)<br>
\( 18x +6y = 42 \)</p>
<p>Let’s subtract the equations:</p>
<p>\( \begin{array}{rcl}18x +6y &=& -48 \\ 18x +6y &=& 42 \\ 0 &=& -90\end{array} \)</p>
<p>Since this equation can never be true, the system has no solutions.</p>
<p>Using another method, you can also write each equation in slope-intercept form:</p>
<p>\( y = -3x - 8 \)<br>
\( y = -3x - 7 \)</p>
<p>Since the equations have the same slope and different \(y\)-intercepts, we know that the system has no solution.</p>
</p>