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<h4>Activity</h4>
<br>
<div class="os-raise-familysupport">
<p>Access the <a href="https://k12.openstax.org/contents/raise/resources/ad776e2f4b03a08bf69ddddc59a1efd31551b1a0" target="_blank">Desmos guide PDF</a> for tips on solving problems with the Desmos graphing calculator.</p>
</div>
<br>
<p>Below are three systems of equations you saw earlier.</p>
<p>Use System A to answer questions 1–6.</p>
<p>System A</p>
<p>\(\left\{\begin{array}{c}4x+3y=10\;\\-4x+5y=6\end{array}\right.\)</p>
<br>
<!--Q#1-->
<ol class="os-raise-noindent">
<li>Use the graphing tool or technology outside the course. Graph the equations for System A that represent this scenario using the Desmos tool below.</li>
</ol>
<div class="os-raise-ib-desmos-gc" data-schema-version="1.0"></div>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<!--QUESTION IS OUTSIDE OF INTERACTION DUE TO DESMOS-->
</div>
<div class="os-raise-ib-cta-prompt">
<p>When you have finished graphing, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal1">
<p>Compare your answer:</p>
<img height="304" src="https://k12.openstax.org/contents/raise/resources/71e8c2e76c0c148e09a4857b0ddf62c81ee77134" width="300"> <br>
<br>
</div>
<!--Interaction End -->
<br>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="62dc0df0-9fae-46f5-9856-b93fdaefafde" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q#2-->
<div class="os-raise-ib-pset-problem" data-content-id="4da44566-324d-4927-8c3f-81dbeb4e1863" data-problem-comparator="integer" data-problem-type="input" data-solution="1">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="2">
<li> Identify the \(x\) coordinate of the solution. </li>
</ol>
<p>Enter your answer here.</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! The answer is 1.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The answer is 1.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q#3-->
<div class="os-raise-ib-pset-problem" data-content-id="efc3fc12-dbd8-4bf8-9e76-6c35c06b71c5" data-problem-comparator="float" data-problem-type="input" data-solution="2">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="3">
<li> Identify the \(y\) coordinate of the solution. </li>
</ol>
<p>Enter your answer here.</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! The answer is 1.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The answer is 1.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Do not edit below line.-->
<div class="os-raise-ib-pset-correct-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-pset-encourage-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<br>
<!--Q#4-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="2635d14d-f295-47e5-8ac4-38c5ee46accb" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="4">
<li> Find the sum or difference of the two original equations that would enable the system to be solved. </li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>Your answer may vary, but here are some samples.</p>
<ul>
<li> Sample Answer 1: \(8y=16 \)</li>
<li> Sample Answer 2: \(y=2 \)</li>
<li> Sample Answer 3: \(8x−2y=4 \)</li>
</ul>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Q#5-->
<ol class="os-raise-noindent" start="5">
<li> Use the graphing tool or technology outside the course. Graph your third equation and the System A equations using the Desmos tool below. </li>
</ol>
<div class="os-raise-ib-desmos-gc" data-schema-version="1.0"></div>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal5" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<!--QUESTION IS OUTSIDE OF INTERACTION DUE TO DESMOS-->
</div>
<div class="os-raise-ib-cta-prompt">
<p>When you have finished graphing, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal5">
<p>Compare your answer:</p>
<p><img alt="Graph of a linear system." src="https://k12.openstax.org/contents/raise/resources/8a942e0e69c4e77278807606033a9fd90613bb3b"></p>
</div>
<!--Interaction End -->
<br>
<br>
<!--Q#6-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="6cd326b5-34fd-45f5-aa52-0c57ed045c3b" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="6">
<li> Make an observation about the graph. </li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>Your answer may vary, but here is a sample.</p>
<p>All three lines intersect at \((1, 2)\). If students graph \(8y=16\), the graph of the new equation is a horizontal line. If students graph \(8x−2y=4\), the graph of the new equation is a new line between the two original lines.</p>
</div>
</div>
<!--Interaction End -->
<br>
<p>Use System B to answer questions 7–12.</p>
<p>System B</p>
<p>\(\left\{\begin{array}{c}2x+y=4\\x-y=11\end{array}\right.\)</p>
<br>
<!--Q#7-->
<ol class="os-raise-noindent" start="7">
<li>Use the graphing tool or technology outside the course. Graph the equations for System B that represent this scenario using the Desmos tool below.</li>
</ol>
<div class="os-raise-ib-desmos-gc" data-schema-version="1.0"></div>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal7" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<!--QUESTION IS OUTSIDE OF INTERACTION DUE TO DESMOS-->
</div>
<div class="os-raise-ib-cta-prompt">
<p>When you have finished graphing, then select the <strong>solution</strong> button to compare your work.
</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal7">
<p>Compare your answer:</p>
<p><img alt="Graph of a linear system." src="https://k12.openstax.org/contents/raise/resources/07785e38d47bfd086d792491687c0f1381c09c29" width="300"></p>
</div>
<!--Interaction End -->
<br>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="ab29034b-98be-42de-a382-a2ba9024301d" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q#8-->
<div class="os-raise-ib-pset-problem" data-content-id="7931bfa9-6a91-45e6-9178-17c48e4825a5" data-problem-comparator="integer" data-problem-type="input" data-solution="5">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="8">
<li> Identify the \(x\) coordinate of the solution. </li>
</ol>
<p>Enter your answer here.</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! The answer is 5. </p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The answer is 5. </p>
</div>
</div>
<!--END QUESTION.-->
<!--Q#9-->
<div class="os-raise-ib-pset-problem" data-content-id="a3dfa06c-07eb-4022-9ead-e3e1ead6d268" data-problem-comparator="float" data-problem-type="input" data-solution="-6">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="9">
<li> Identify the \(y\) coordinate of the solution. </li>
</ol>
<p>Enter your answer here.</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! The answer is -6. </p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The answer is -6. </p>
</div>
</div>
<!--END QUESTION.-->
<!--Do not edit below line.-->
<div class="os-raise-ib-pset-correct-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-pset-encourage-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<br>
<!--Q#10-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="3eb4ea38-16d6-4e4d-84cb-3adfb0693072" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="10">
<li> Find the sum or difference of the two original equations that would enable the system to be solved. </li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>Your answer may vary, but here are some samples.</p>
<ul>
<li> Sample Answer 1: \(3x=15 \)</li>
<li> Sample Answer 2: \(x=5 \)</li>
</ul>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Q#11-->
<ol class="os-raise-noindent" start="11">
<li>Use the graphing tool or technology outside the course. Graph your third equation and the System B equations using the Desmos tool below. </li>
</ol>
<div class="os-raise-ib-desmos-gc" data-schema-version="1.0"></div>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal11" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<!--QUESTION IS OUTSIDE OF INTERACTION DUE TO DESMOS-->
</div>
<div class="os-raise-ib-cta-prompt">
<p>When you have finished graphing, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal11">
<p>Compare your answer:</p>
<p><img alt="Graph of a linear system." src="https://k12.openstax.org/contents/raise/resources/4791d4d27317a14a058f9373881c4584c9e4669a"></p>
</div>
<!--Interaction End -->
<br>
<br>
<!--Q#12-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="2b17a41c-b310-4de6-843d-0f53a65afb33" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="12">
<li> Make an observation about the graph. </li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>Your answer may vary, but here is a sample. All three lines intersect at \((5, -6)\). If students graph \(3x=15\), the graph of the new equation is a vertical line.</p>
</div>
</div>
<!--Interaction End -->
<br>
<p>Use System C to answer questions 13–18.</p>
<p>System C</p>
<p>\(\left\{\begin{array}{c}8x+11y=37\\8x+y=7\end{array}\right.\)</p>
<br>
<!--Q13-->
<ol class="os-raise-noindent" start="13">
<li> Use the graphing tool or technology outside the course. Graph the equations for System C that represent this scenario using the Desmos tool below. </li>
</ol>
<div class="os-raise-ib-desmos-gc" data-schema-version="1.0"></div>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal13" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<!--QUESTION IS OUTSIDE OF INTERACTION DUE TO DESMOS-->
</div>
<div class="os-raise-ib-cta-prompt">
<p>When you have finished graphing, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal13">
<p>Compare your answer:</p>
<p><img alt="Graph of a linear system." src="https://k12.openstax.org/contents/raise/resources/44ebd2bb4d629f315f8a352bf375bb4d8ff9b12e" width="300"></p>
</div>
<!--Interaction End -->
<br>
<br>
<!--Q#14-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="00fc6a24-434e-45ef-a8b6-7161bac92ef4" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="14">
<li> Identify the \(x\) coordinate of the solution. </li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>\(\frac12\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="844062ac-6210-4ddb-a528-58b885d11b3c" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0"><!--Q#15-->
<div class="os-raise-ib-pset-problem" data-content-id="f55ee52c-860b-453e-bf06-d178b89a8990" data-problem-comparator="integer" data-problem-type="input" data-solution="3">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="15">
<li> Identify the \(y\) coordinate of the solution. </li>
</ol>
<p>Enter your answer here.</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! The answer is 3. </p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The answer is 3. </p>
</div>
</div>
<!--END QUESTION.-->
<!--Do not edit below line.-->
<div class="os-raise-ib-pset-correct-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-pset-encourage-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<br>
<!--Q#16-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="cc9cb709-873e-4b28-b6d7-01573f8aee7e" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="16">
<li> Find the sum or difference of the two original equations that would enable the system to be solved. </li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>Your answer may vary, but here are some samples.</p>
<ul>
<li> Sample Answer 1: \(10y=30 \)</li>
<li> Sample Answer 2: \(−10y=−30 \)</li>
<li> Sample Answer 3: \(y=3 \)</li>
<li> Sample Answer 4: (If the equations are added) \(16x+12y=44 \)</li>
</ul>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Q#17-->
<ol class="os-raise-noindent" start="17">
<li> Use the graphing tool or technology outside the course. Graph your third equation and the System C equations using the Desmos tool below. </li>
</ol>
<div class="os-raise-ib-desmos-gc" data-schema-version="1.0"></div>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal17" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<!--QUESTION IS OUTSIDE OF INTERACTION DUE TO DESMOS-->
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal17">
<p>Compare your answer:</p>
<p><img alt="Graph of a linear system.Graph of a linear system." src="https://k12.openstax.org/contents/raise/resources/680c69a559ee7e9bd444ef9bd3adf1cd1e955390"></p>
</div>
<!--Interaction End -->
<br>
<br>
<!--Q#18-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="568830f8-8694-4f89-8092-21077da5cae3" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="18">
<li> Make an observation about the graph. </li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>Your answer may vary, but here is a sample. All three lines intersect at \((\frac12,3)\). If students graph \(10y=30\), the graph of the new equation is a horizontal line.</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!-- "Start "Are you Ready for More? click to reveal. If you have more than one on a page, you will need to change the Data-fire/data-wait-for-events for each set.-->
<div class="os-raise-ib-cta" data-button-text="Are you ready for more?" data-fire-event="RFM1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-cta-prompt">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<!--Start interaction. If multiple interactions appear under "are you ready for more?, they should all have matching "data-wait-for-event", which should also match the "data-fire-event" for the button. Note in this sample, they are all RFM1-->
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<h4>Extending Your Thinking</h4>
<p>Mai wonders what would happen if we multiply equations. That is, we multiply the expressions on the left side of the two equations and set them equal to the expressions on the right side of the two equations.</p>
<p>Use System B to answer questions 1–4.</p>
<p>System B</p>
<p>\(\left\{\begin{array}{c}2x+y=4\\x-y=11\end{array}\right.\)</p>
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<li> Write out an equation that you would get if you multiply the two equations in this manner. </li>
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<p>Enter your answer here:</p>
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<p>Compare your answer:</p>
<p>Your answer may vary, but here is a sample. \((2x+y)(x−y)=44\)</p>
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<li> Does your original solution still work in this new equation? </li>
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<p>Enter your answer here:</p>
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<p>Compare your answer:</p>
<p>Yes</p>
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<li> Use the graphing tool or technology outside the course. Graph your third equation and the System B equations using the Desmos tool below. </li>
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<p>When you have finished graphing, then select the <strong>solution</strong> button to compare your work.</p>
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<p>Compare your answer:</p>
<p><img alt="A system of equations." src="https://k12.openstax.org/contents/raise/resources/efaaa21ce115d24d626d1fbe16deb365977e88c4"></p>
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<li> Explain if the approach of graphing the new equation is or is not particularly helpful. </li>
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<p>Enter your answer here:</p>
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<p>Compare your answer:</p>
<p>This approach is not helpful because the new equation is not a line and therefore not so easy to work with.</p>
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