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ebcc588b-442e-4b21-ba18-e8b1b0adef7a.html
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<h4>Activity</h4>
<p>With a partner, discuss the following situations and equations. For each, consider:</p>
<ul>
<li>
What does the solution mean in the context of the situation?
</li>
<li>
Are the given values solutions?
</li>
</ul>
<p>If operations are applied correctly, the solution to an equation is also the solution to all equations equivalent to
it. If operations are incorrectly applied, the solution of each equation is different.</p>
<p>For numbers 1 and 2, use the following situation and equation:</p>
<p>Noah is buying a pair of jeans and using a coupon for 10% off. The total price is $56.70, which includes $2.70 in
sales tax. Noah’s purchase can be modeled by the equation:</p>
<p>\(x−0.1x+2.70=56.70\)</p>
<!--Q1-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="1d3feb9b-c77a-481c-b3ce-8f1638a8f870" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<li>
What does the solution to the equation mean in this situation?
</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.<br>
The solution is the price of a pair of jeans.</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Q2-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="caf0f3a0-abee-4b19-ba09-e0e70be1ef41" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2">
<li>
How can you verify that 70 is not a solution but 60 is 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>Your answer may vary, but here are some samples.</p>
<p>70 can be substituted into the equation to make a false statement:<br>
\(70−0.1(70)+2.70=56.70\)<br>
\(65.70 ≠ 56.70\) ❌</p>
<p>60 can be substituted into the equation to make a true statement:<br>
\(
60−0.1(60)+2.70=56.70\)<br>
\(56.70=56.70\) \(\checkmark\) </p>
<p>Here are some equations that are related to \(x−0.1x+2.70=56.70\). Each equation is a result of performing one or
more moves on that original equation. Each can also be interpreted in terms of Noah’s purchase.</p>
<p>For numbers 3–5, use the equation \(100x−10x+270=5,670\).</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Q3-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="4a7ff54a-ff9f-4d06-bfc6-88da1afa9932" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="3">
<li>
What was done to Noah’s equation to make this one?
</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.<br>
Every term was multiplied by 100. The equation is equivalent to Noah’s equation.</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Q4-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="159965c1-ba48-4e5e-b403-019edf9d6ca5" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="4">
<li>
What is the interpretation of this new equation?
</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.<br>
The price is now expressed in cents instead of dollars.</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Q5-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="aa190d54-4702-4bfb-b84f-b80ed1231577" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="5">
<li>
Is the solution the same?
</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.<br>
60 is a solution in both equations.</p>
<p>For numbers 6–8, use the equation \(x−0.1x=54\).</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Q6-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="35217e55-2288-45f2-a9e3-8d8dea57308d" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="6">
<li>
What was done to Noah’s equation to make this one?
</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.<br>
Subtract 2.70 from both sides of the equation. The equation is equivalent to Noah’s equation.</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Q7-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="9fd9c1e2-e719-49cc-9a87-73971faf79d7" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="7">
<li>
What is the interpretation of this new equation?
</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.<br>
The price is still in dollars.</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Q8-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="072b0b1f-0292-458a-a701-c31ebeba59a4" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="8">
<li>
Is the solution the same?
</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.<br>
60 is a solution in both equations.</p>
<p>For numbers 9–11, use the equation \(0.9x+2.70=56.70\).</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Q9-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="1a593b74-07be-49a0-b5eb-27362f743654" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="9">
<li>
What was done to Noah’s equation to make this one?
</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.<br>
The terms \(x\) and 0.1x were correctly combined to create an equivalent equation to Noah’s original equation.
</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Q10-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="e78be93f-5003-44c3-9a22-b127d5bfa612" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="10">
<li>
What is the interpretation of this new equation?
</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>10% off means paying 90% of the original price. 90% of the original price plus sales tax is $56.70.</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Q11-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="18ad49c7-c877-402d-a27c-115b90c0715a" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="11">
<li>
Is the solution the same?
</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>60 is a solution in both equations.</p>
</div>
</div>
<!--Interaction End -->
<br>
<p>For numbers 12–14, use the equation \(x−0.1x=56.70\).</p>
<br>
<!--Q12-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="2c2f723f-a73f-4b78-acb8-a03e27bd4c10" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="12">
<li>
What was done to Noah’s equation to make this one?
</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>Your answer may vary, but here is a sample.</p>
<p>Sales tax was not included on the left but was included on the right. This equation is not equivalent to Noah’s
original equation.</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Q13-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="9aa9337d-8151-4fe3-932a-36b776619e33" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="13">
<li>
What is the interpretation of this new equation?
</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>Your answer may vary, but here is a sample.</p>
<p>The price after using the coupon for 10% off before sales tax is not the same as the price after using the coupon for
10% off including the sales tax, $56.70.</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Q14-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="1a3b4709-a942-4282-b269-9c3c5f558704" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="14">
<li>
Is the solution the same?
</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>Your answer may vary, but here is a sample.</p>
<p>60 is not a solution to this equation.</p>
</div>
</div>
<!--Interaction End -->
<br>
<p>For numbers 15–17, use the equation \(x−0.1x=59.40\)</p>
<br>
<!--Q15-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="b2d8cf20-4ad5-4e32-a5b7-f6c8afb0b470" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="15">
<li>
What was done to Noah’s equation to make this one?
</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>Your answer may vary, but here is a sample.<br>
Subtract 2.70 from the left and add 2.70 to the right.</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Q16-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="05d0648b-19cf-4302-98f3-422b40a3fa97" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="16">
<li>
What is the interpretation of this new equation?
</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>Your answer may vary, but here is a sample.</p>
<p>The same operation must be done to both sides of an equation to keep it balanced and to create an equivalent
equation. The price after using the coupon for 10% off is not equal to 56.70 plus tax.</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Q17-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="543e17d8-a0aa-4a2d-a643-b7cdbf8db044" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="17">
<li>
Is the solution the same?
</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>Your answer may vary, but here is a sample.<br>
60 is not a solution to this equation.</p>
</div>
</div>
<!--Interaction End -->
<br>
<p>For numbers 18–20, use the equation \(2(x−0.1x+2.70)=56.70\).</p>
<br>
<!--Q18-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="20f95f59-977c-4344-b6f3-4b44b98dc08b" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="18">
<li>What was done to Noah’s equation to make this one?</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>Your answer may vary, but here is a sample.<br>
The left side only was multiplied by 2.</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Q19-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="7d01610b-2b6e-48ee-a01d-b809d3c0172a" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="19">
<li>What is the interpretation of this new equation?</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>Your answer may vary, but here is a sample.<br>
Two pairs of jeans after the 10% discount and sales tax cost 56.70.</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Q20-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="6a74ec0e-a850-45e1-945d-2b2c282f1f52" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="20">
<li>Is the solution the same?</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>Your answer may vary, but here is a sample.<br>
60 is not a solution to this equation.</p>
</div>
</div>
<!--Interaction End -->
<br>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="07b1abb7-c8c6-4dd8-bd49-9f6938d2f87e" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q21-->
<div class="os-raise-ib-pset-problem" data-content-id="9f003b7b-c462-41fb-9132-5dfd019c4547" data-problem-type="multiselect" data-solution='["\\(100x-10x+270=5670\\)", "\\(x-0.1x=54\\)", "\\(0.9x+2.70=56.70\\)"]' data-solution-options='["\\(100x-10x+270=5670\\)", "\\(x-0.1x=54\\)", "\\(0.9x+2.70=56.70\\)","\\(x-0.1x=56.70\\)", "\\(x-0.1x=59.40\\)", "\\(2(x-0.1x+2.70)=56.70\\)"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="21">
<li>Based on your work above, which of the six equations are equivalent to the original equation, \(x−0.1x+2.70=56.70\)? Select the <strong>three</strong> equations that are equivalent to the original.</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Compare your answer: A, B, C were all equivalent equations to the original with 60 as a solution.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answers are \(100x-10x+270=5670\), \(x-0.1x=54\), and \(0.9x+2.70=56.70\). </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>
<h4>Video: Looking at Equivalent Equations</h4>
<p>Watch the following video to learn more about why these are equivalent equations.</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/337b23511ca7ac0eb4d32eba29110835c40350e5">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/4ee789e942e6571879491bca1b89dac5cb04a66a " srclang="en_us">
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