Thanks a lot... So multiply by the reciprocal of the fraction is the necessary way to get rid of the fraction from the variable x . This is interesting 

<h2>Solving Equations Using the Division and Multiplication Properties of Equality</h2>
<p>We introduced the Multiplication and Division Properties of Equality in <a data-page="19" href="https://app.ulearngo.com/tutorial/mathematics-103/">Mathematics 103</a> and <a data-page="27" href="https://app.ulearngo.com/tutorial/mathematics-104/">Mathematics 104</a>. We modeled how these properties worked using envelopes and counters and then applied them to solving equations (See <a data-page="19" href="https://app.ulearngo.com/tutorial/mathematics-103/">Mathematics 103</a>). We restate them again here as we prepare to use these properties again.</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Definition: Division and Multiplication Properties of Equality</h3>
<p><strong>Division Property of Equality</strong>: For all real numbers \(a,b,c,\) and \(c\ne 0,\) if \(a=b,\) then \(\frac{a}{c}=\frac{b}{c}.\)</p>
<p><strong>Multiplication Property of Equality</strong>: For all real numbers \(a,b,c,\) if \(a=b,\) then \(ac=bc.\)</p>
</div>
</div>
<p>When you divide or multiply both sides of an equation by the same quantity, you still have equality.</p>
<p>Let’s review how these properties of equality can be applied in order to solve equations. Remember, the goal is to ‘undo’ the operation on the variable. In the example below the variable is multiplied by \(4,\) so we will divide both sides by \(4\) to ‘undo’ the multiplication.</p>
<h3>Optional Video: Solving One Step Equation by Mult/Div. Integers (Var on Left)</h3>
<p><iframe allowfullscreen="allowfullscreen" frameborder="0" height="337" src="https://www.youtube.com/embed/rZuvbYO3sV8?rel=0&amp;modestbranding=1&amp;cc_lang_pref=en&amp;cc_load_policy=1" width="599"></iframe></p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Solve: \(4x=-28.\)</p>
</div>
<div>
<h3>Solution</h3>
<p>We use the Division Property of Equality to divide both sides by \(4.\)</p>
<table data-label="" style="width: 497px;" summary="The top line shows 4x equals negative 28. The next line says “Divide both sides by 4 to undo the multiplication” and shows 4x over a red 4 equals negative 28 over a red 4. The next step says “Simplify” and shows x equal negative seven. The next line says “Check your answer. Let x equal negative seven.” and shows 4x equals negative 28. The next line shows four times negative seven may be equal to negative twenty-eight. The last line shows that negative twenty-eight equals negative twenty-eight with a check mark.">
<tbody>
<tr>
<td style="width: 349px;"> </td>
<td style="width: 134px;"><img alt="." src="https://data.ulearngo.com/assets/ccea5045-e0ae-4e9c-bf12-c0a936216974"/></td>
</tr>
<tr>
<td style="width: 349px;">Divide both sides by 4 to undo the multiplication.</td>
<td style="width: 134px;"><img alt="." src="https://data.ulearngo.com/assets/47db4ec6-ee95-454b-94a6-7d47e3b06846"/></td>
</tr>
<tr>
<td style="width: 349px;">Simplify.</td>
<td style="width: 134px;"><img alt="." src="https://data.ulearngo.com/assets/b21290b4-377e-4ec6-b4fd-28cbb614ca4f"/></td>
</tr>
<tr>
<td style="width: 349px;">Check your answer. Let \(x=-7\).</td>
<td style="width: 134px;"> </td>
</tr>
<tr>
<td style="width: 349px;"><img alt="." src="https://data.ulearngo.com/assets/70c7ba13-e9cb-453b-9ed8-11c2d3a52d5f"/></td>
<td style="width: 134px;"> </td>
</tr>
<tr>
<td style="width: 349px;"><img alt="." src="https://data.ulearngo.com/assets/5d00afdf-4d6d-4221-a526-9c3cedfa1b28"/></td>
<td style="width: 134px;"> </td>
</tr>
<tr>
<td style="width: 349px;"><img alt="." src="https://data.ulearngo.com/assets/d15788bc-1ca1-4787-a2f0-6d9380aa7171"/></td>
<td style="width: 134px;"> </td>
</tr>
</tbody>
</table>
<p>Since this is a true statement, \(x=-7\) is a solution to \(4x=-28.\)</p>
</div>
</div>
</div>
</div>
</div>
<p>In the previous example, to ‘undo’ multiplication, we divided. How do you think we ‘undo’ division?</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Solve: \(\frac{\phantom{\rule{0.4em}{0ex}}a}{-7}=-42.\)</p>
</div>
<div>
<h3>Solution</h3>
<p>Here \(a\) is divided by \(-7.\) We can multiply both sides by \(-7\) to isolate \(a.\)</p>
<table data-label="" style="width: 497px;" summary="The top shows a over negative 7 equals negative 42. The next line says “Multiply both sides by negative 7” and shows a red negative 7 times a over negative 7 equals a red negative 7 times negative 42. The next step shows negative 7a over negative 7 equals 294. The next line says “Simplify” and shows a equals 294. The next line says “Check your answer. Let a equal 294.” and shows a over negative 7 equals negative 42. The next line shows 294 over negative seven may be equal to negative 42. The last line shows that negative 42 equals negative 42 with a check mark.">
<tbody>
<tr>
<td style="width: 307px;"> </td>
<td style="width: 176px;"><img alt="." src="https://data.ulearngo.com/assets/f3f76941-6491-41d7-8222-19f39c0912fb"/></td>
</tr>
<tr>
<td style="width: 307px;">Multiply both sides by \(-7\).</td>
<td style="width: 176px;"><img alt="." src="https://data.ulearngo.com/assets/fc5afb9d-d06e-4e50-892f-d4a95246ff1b"/>
<div> </div>
<img alt="." src="https://data.ulearngo.com/assets/bd7d3010-e801-4a78-b187-2a14dc61fb35"/></td>
</tr>
<tr>
<td style="width: 307px;">Simplify.</td>
<td style="width: 176px;"><img alt="." src="https://data.ulearngo.com/assets/96e633aa-7754-4c36-9eb6-440505afdc2b"/></td>
</tr>
<tr>
<td style="width: 307px;">Check your answer. Let \(a=294\).</td>
<td style="width: 176px;"> </td>
</tr>
<tr>
<td style="width: 307px;"><img alt="." src="https://data.ulearngo.com/assets/3f9b293f-5a2a-4fcd-935b-649c912970e5"/></td>
<td style="width: 176px;"> </td>
</tr>
<tr>
<td style="width: 307px;"><img alt="." src="https://data.ulearngo.com/assets/329e8fd0-60cb-4ac8-ae37-c63d19895fa4"/></td>
<td style="width: 176px;"> </td>
</tr>
<tr>
<td style="width: 307px;"><img alt="." src="https://data.ulearngo.com/assets/4dafb332-b5fd-4cdd-873d-163e4a06a99a"/></td>
<td style="width: 176px;"> </td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Solve: \(-r=2.\)</p>
</div>
<div>
<h3>Solution</h3>
<p>Remember \(-r\) is equivalent to \(-1r.\)</p>
<table data-label="" style="width: 436px;" summary="The first line says negative r equals 2. The next line says “Rewrite negative r as negative 1 times r” and shows negative 1r equals 2. The next line says “Divide both sides by negative 1” and shows negative 1r over red negative 1 equals 2 over red negative 1. The next line says to divide and shows r equals negative 2. The next line says to check and shows negative r equals 2. It then says to substitute r equals negative 2 and shows negative red negative 2 followed by an equal sign with a question mark and then a 2. The last line says to simplify and shows that 2 equals 2, followed by a check mark.">
<tbody>
<tr>
<td colspan="2" style="width: 299px;"> </td>
<td style="width: 119px;"><img alt="." src="https://data.ulearngo.com/assets/849493fe-1346-4ccd-a78d-d703e881e02d"/></td>
</tr>
<tr>
<td colspan="2" style="width: 299px;">Rewrite \(-r\) as \(-1r\).</td>
<td style="width: 119px;"><img alt="." src="https://data.ulearngo.com/assets/6b351144-e74b-4c5f-980b-9d93311e79de"/></td>
</tr>
<tr>
<td colspan="2" style="width: 299px;">Divide both sides by \(-1\).</td>
<td style="width: 119px;"><img alt="." src="https://data.ulearngo.com/assets/15ea663f-7fee-4a07-a943-217d9a2986c3"/></td>
</tr>
<tr>
<td colspan="2" style="width: 299px;"> </td>
<td style="width: 119px;"><img alt="." src="https://data.ulearngo.com/assets/624041d7-52d2-478b-a089-759b2dedddb4"/></td>
</tr>
<tr>
<td style="width: 176px;">Check.</td>
<td style="width: 123px;"><img alt="." src="https://data.ulearngo.com/assets/b9b6b017-8ea1-44f1-8f0d-7b9d593f75cc"/></td>
<td style="width: 119px;"> </td>
</tr>
<tr>
<td style="width: 176px;">Substitute \(r=-2\)</td>
<td style="width: 123px;"><img alt="." src="https://data.ulearngo.com/assets/bc4e5638-4fb3-4d1f-87fe-2f991558939e"/></td>
<td style="width: 119px;"> </td>
</tr>
<tr>
<td style="width: 176px;">Simplify.</td>
<td style="width: 123px;"><img alt="." src="https://data.ulearngo.com/assets/6786b61c-0cbd-4a4b-9453-dd8615cc0d48"/></td>
<td style="width: 119px;"> </td>
</tr>
</tbody>
</table>
<p>In <a data-page="27" href="https://app.ulearngo.com/tutorial/mathematics-104/">Mathematics 104</a>, we saw that there are two other ways to solve \(-r=2.\)</p>
<p>We could multiply both sides by \(-1.\)</p>
<p>We could take the opposite of both sides.</p>
</div>
</div>
</div>
</div>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Solve: \(\frac{2}{3}\phantom{\rule{0.1em}{0ex}}x=18.\)</p>
</div>
<div>
<h3>Solution</h3>
<p>Since the product of a number and its reciprocal is \(1,\) our strategy will be to isolate \(x\) by multiplying by the reciprocal of \(\frac{2}{3}.\)</p>
<table data-label="" style="width: 538px;" summary="The first line shows two-thirds x equals 18. The next line says “Multiply by the reciprocal of two-thirds” and shows a red 3 over 2 times two-thirds x equals a red 3 over 2 times 18. The next line says “Reciprocals multiply to 1” and shows 1x equals 3 over 2 times 18 over 1. The next line says “Multiply” and shows x equals 27. The next line says “Check your answer. Let x equal 27 and shows two-thirds x equals 18. The next line shows two-thirds times 27 may equal 18. The last line shows that 18 equals 18 with a check mark.">
<tbody>
<tr>
<td style="width: 367px;"> </td>
<td style="width: 157px;"><img alt="." src="https://data.ulearngo.com/assets/addadcde-e131-4449-af7e-7e4a7113381e"/></td>
</tr>
<tr>
<td style="width: 367px;">Multiply by the reciprocal of \(\frac{2}{3}\).</td>
<td style="width: 157px;"><img alt="." src="https://data.ulearngo.com/assets/097684b9-422f-453d-9139-789b3d24db73"/></td>
</tr>
<tr>
<td style="width: 367px;">Reciprocals multiply to one.</td>
<td style="width: 157px;"><img alt="." src="https://data.ulearngo.com/assets/1f57ae00-68fc-42a8-96b3-a4b4bbc8975d"/></td>
</tr>
<tr>
<td style="width: 367px;">Multiply.</td>
<td style="width: 157px;"><img alt="." src="https://data.ulearngo.com/assets/ddbacd87-81b8-408c-98fa-a3a2c08897d8"/></td>
</tr>
<tr>
<td style="width: 367px;">Check your answer. Let \(x=27\)</td>
<td style="width: 157px;"> </td>
</tr>
<tr>
<td style="width: 367px;"><img alt="." src="https://data.ulearngo.com/assets/6b4eb4cd-67f6-4834-955b-e71d26b4b82b"/></td>
<td style="width: 157px;"> </td>
</tr>
<tr>
<td style="width: 367px;"><img alt="." src="https://data.ulearngo.com/assets/f1c394d8-539d-4949-a234-d4ad4d4e007f"/></td>
<td style="width: 157px;"> </td>
</tr>
<tr>
<td style="width: 367px;"><img alt="." src="https://data.ulearngo.com/assets/d6e0510e-9eac-4d81-b7c3-f5b0d902ad27"/></td>
<td style="width: 157px;"> </td>
</tr>
</tbody>
</table>
<p>Notice that we could have divided both sides of the equation \(\frac{2}{3}\phantom{\rule{0.1em}{0ex}}x=18\) by \(\frac{2}{3}\) to isolate \(x.\) While this would work, multiplying by the reciprocal requires fewer steps.</p>
</div>
</div>
</div>
</div>
</div>
<h3>Optional Video: Solving One Step Equation by Mult/Div. Integers (Var on Right)</h3>
<p><iframe allowfullscreen="allowfullscreen" frameborder="0" height="337" src="https://www.youtube.com/embed/TB1rkPbF8rA?rel=0&amp;modestbranding=1&amp;cc_lang_pref=en&amp;cc_load_policy=1" width="599"><span class="mce_SELRES_start" data-mce-type="bookmark" style="display: inline-block; width: 0px; overflow: hidden; line-height: 0;">﻿</span></iframe></p>

77e14597-4d58-4399-be3f-b1bb5c28bc54

Solving Equations Using the Division and Multiplication Properties of Equality

Mathematics studies measurement, relationships, and properties of quantities and sets, using numbers and symbols. Arithmetic, algebra, geometry, and calculus are popular topics in mathematics.

Mathematics

Discover how to solve linear equations using various properties of equality, simplifying equations, translation and solving of equations, and a general strategy approach for equations with fraction and decimal coefficients.


Solving Equations Using the Division and Multiplication Properties of Equality

Lesson video

Lesson video

Track Your Learning Progress

Earlier discussion