<h2>Solving Equations Using the Division and Multiplication Properties of Equality</h2>
<p>You may have noticed that all of the equations we have solved so far have been of the form \(x+a=b\) or \(x-a=b\). We were able to isolate the variable by adding or subtracting the constant term on the side of the equation with the variable. Now we will see how to solve equations that have a variable multiplied by a constant and so will require division to isolate the variable.</p>
<p>Let’s look at our puzzle again with the envelopes and counters in the figure below.</p>
<div class="wp-caption alignnone" style="width: 215px"><img alt="This image illustrates a workspace divided into two sides. The content of the left side is equal to the content of the right side. On the left side, there are two envelopes each containing an unknown but equal number of counters. On the right side are six counters." height="163" src="https://data.ulearngo.com/assets/ea8274b5-0284-4d32-8401-1ed28395e3a0" width="215"/><p class="wp-caption-text">The illustration shows a model of an equation with one variable multiplied by a constant. On the left side of the workspace are two instances of the unknown (envelope), while on the right side of the workspace are six counters.</p></div>
<p>In the illustration there are two identical envelopes that contain the same number of counters. Remember, the left side of the workspace must equal the right side, but the counters on the left side are “hidden” in the envelopes. So how many counters are in each envelope?</p>
<p>How do we determine the number? We have to separate the counters on the right side into two groups of the same size to correspond with the two envelopes on the left side. The 6 counters divided into 2 equal groups gives 3 counters in each group (since \(6÷2=3\)).</p>
<p>What equation models the situation shown in the figure below? There are two envelopes, and each contains \(x\) counters. Together, the two envelopes must contain a total of 6 counters.</p>
<div class="wp-caption alignnone" style="width: 211px"><img alt="This image illustrates a workspace divided into two sides. The content of the left side is equal to the content of the right side. On the left side, there are two envelopes each containing an unknown but equal number of counters. On the right side are six counters. Underneath the image is the equation modeled by the counters: 2 x equals 6." height="193" src="https://data.ulearngo.com/assets/de4e49fa-1343-49d7-8271-aa9fcb086602" width="211"/><p class="wp-caption-text">The illustration shows a model of the equation \(2x=6\).</p></div>
<table data-label="" summary="This figure shows the steps for solving the equation 2x equals 6. The figure has two columns, with written instructions on the left and math on the right. At the top of the figure on the right is the equation 2 x equals 6. One line down on the left, the instructions say: “If we divide both sides of the equation by 2, as we did with the envelopes and counters, we get”. To the right of this phrase is the same equation divided by 2 on both sides: 2 x over 2 equals 6 over 2, with divided by 2 written in red on both sides. Another line down to the right is the answer to the equation: x equals 3.">
<tbody>
<tr>
<td> </td>
<td><img alt="." src="https://data.ulearngo.com/assets/dde07921-7839-4545-8cb3-a8107df52c92"/></td>
</tr>
<tr>
<td data-valign="top">If we divide both sides of the equation by 2, as we did with the envelopes and counters,</td>
<td data-valign="top"><img alt="." src="https://data.ulearngo.com/assets/ea08ba69-b1df-495a-a833-d23e56def0f1"/></td>
</tr>
<tr>
<td data-valign="top">we get:</td>
<td data-valign="top"><img alt="." src="https://data.ulearngo.com/assets/5e7a86fc-2e67-45ef-9cb6-0d4a0cc475bc"/></td>
</tr>
</tbody>
</table>
<p>We found that each envelope contains 3 counters. Does this check? We know \(2·3=6\), so it works! Three counters in each of two envelopes does equal six!</p>
<p>This example leads to the <strong>Division Property of Equality</strong>.</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>The Division Property of Equality</h3>
<p>For any numbers <em>a</em>, <em>b</em>, and <em>c</em>, and \(c\ne 0\),</p>
<div data-label="">\(\begin{array}{ccccc}\text{If}\hfill &amp; &amp; \hfill a&amp; =\hfill &amp; b,\hfill \\ \text{then}\hfill &amp; &amp; \hfill \frac{a}{c}&amp; =\hfill &amp; \frac{b}{c}\hfill \end{array}\)</div>
<p>When you divide both sides of an equation by any non-zero number, you still have equality.</p>
</div>
</div>
<div class="ns-tutorial">
<p>The goal in solving an equation is to ‘undo’ the operation on the variable. In the next example, the variable is multiplied by 5, so we will divide both sides by 5 to ‘undo’ the multiplication.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Solve: \(5x=-27.\)</p>
</div>
<div>
<h3>Solution</h3>
<table data-label="" summary="This figure has two columns, with written instructions on the left and math on the right. At the top of the figure on the left, the instructions say: “To isolate x, ‘undo’ the multiplication by 5.” To the right of this sentence is the equation 5 x equals negative 27. The next line down on the left, the instructions say: “Divide to ‘undo’ the multiplication.” To the right of this sentence is the same equation with both sides divided by 5: 5 x over 5 equals negative 27 over 5, with divided by 5 written in red on both sides. Another line down on the left, the instructions say: “Simplify.” To the right of this sentence is the answer to the equation: x equals negative 27/5. Another line down to the left, the instructions say: Check”. To the right of this instruction is the original equation again: 5x equals negative 27. Another line down, the instructions say to substitute negative 27/5 for x, and to the right is the equation with negative 27/5 substituted for x: 5 times negative 27/5, with negative 27/5 in parentheses and written in red, equals negative 27. Below this is the equation negative 27 equals negative 27, with a check mark next to it.  Another line down, in the right-hand column, is a sentence that reads “Since this is a true statement x = -27/5 is the solution to 5x = -27.”">
<tbody>
<tr>
<td colspan="2">To isolate \(x\), “undo” the multiplication by 5.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/ebb455cb-37c5-4009-b7e6-9ffe6d7ce027"/></td>
</tr>
<tr>
<td colspan="2">Divide to ‘undo’ the multiplication.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/6350f1ce-b3a6-48b4-b6e5-ca391882be26"/></td>
</tr>
<tr>
<td colspan="2">Simplify.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/78f0416d-c147-46e0-b4c4-efe2575dfa24"/></td>
</tr>
<tr>
<td>Check:</td>
<td><img alt="." src="https://data.ulearngo.com/assets/79c31323-0af4-432e-8180-e149d615850d"/></td>
<td> </td>
</tr>
<tr>
<td>Substitute \(-\frac{27}{5}\) for \(x.\)</td>
<td><img alt="." src="https://data.ulearngo.com/assets/d9b9412b-1d7e-47bd-b902-8d89b3bc4852"/></td>
<td> </td>
</tr>
<tr>
<td> </td>
<td><img alt="." src="https://data.ulearngo.com/assets/975bcda2-e857-4c3c-8a0e-617d35aae76a"/></td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td>Since this is a true statement, \(x=-\frac{27}{5}\) is the solution to \(5x=-27\).</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
</div>
</div>
<p>Consider the equation \(\frac{x}{4}=3\). We want to know what number divided by 4 gives 3. So to “undo” the division, we will need to multiply by 4. The <strong>Multiplication Property of Equality</strong> will allow us to do this. This property says that if we start with two equal quantities and multiply both by the same number, the results are equal.</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>The Multiplication Property of Equality</h3>
<p>For any numbers <em>a</em>, <em>b</em>, and <em>c</em>,</p>
<div data-label="">\(\begin{array}{ccccc}\text{If}\hfill &amp; &amp; \hfill a&amp; =\hfill &amp; b,\hfill \\ \text{then}\hfill &amp; &amp; \hfill ac&amp; =\hfill &amp; bc\hfill \end{array}\)</div>
<p>If you multiply both sides of an equation by the same number, you still have equality.</p>
</div>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Solve: \(\frac{y}{-7}=-14.\)</p>
</div>
<div>
<h3>Solution</h3>
<p>Here \(y\) is divided by \(-7\). We must multiply by \(-7\) to isolate \(y\).</p>
<table data-label="" summary="This figure has two columns, with written instructions on the left and math on the right. At the top of the figure on the right is the equation y over negative 7 equals negative 14. One line down on the left, the instructions say: “Multiply both sides by negative 7.” To the right of this sentence is the same equation multiplied by negative 7 on both sides: negative 7 times y over negative 7, with y over negative 7 in parentheses, equals negative 7 times negative 14, with multiplied by negative 7 written in red on both sides. Another line down on the left, the instructions say: “Multiply.” To the right of this sentence is the equation negative 7y over negative 7 equals 98. Another line down to the left, the instructions say: “Simplify.” To the right of this instruction is the answer to the equation: y equals 98. Two lines down, the instructions say: “Check”. To the right of this word is the original equation again: y over negative 7 equals negative 14. Then the instructions say to substitute 98 for y, and to the right is the equation with 98 substituted in for y: 98 (written in red) divided by negative 7 might equal negative 14. Below this the instructions say to divide, and to the right is the equation negative 14 equals negative 14, with a check mark next to it.">
<tbody>
<tr>
<td colspan="2"> </td>
<td><img alt="." src="https://data.ulearngo.com/assets/2927c764-0be6-478e-b528-6d56d7cd9b06"/></td>
</tr>
<tr>
<td colspan="2">Multiply both sides by \(-7\).</td>
<td><img alt="." src="https://data.ulearngo.com/assets/5681d4a2-540f-453f-8a1b-d186f2de3f8a"/></td>
</tr>
<tr>
<td colspan="2">Multiply.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/f330c824-510d-4aef-80c8-31ca664e1e4c"/></td>
</tr>
<tr>
<td colspan="2">Simplify.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/c8bc3e31-c7ee-4064-a059-fb0514ad1eff"/></td>
</tr>
<tr>
<td>Check: \(\frac{y}{-7}=-14\)</td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>Substitute \(y=98\).</td>
<td><img alt="." src="https://data.ulearngo.com/assets/a6296369-2c22-43f6-a98f-00ecc9ca434e"/></td>
<td> </td>
</tr>
<tr>
<td>Divide.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/0fda96e4-77fd-493a-9c1c-f2907f8c0469"/></td>
<td> </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: \(\text{−}n=9.\)</p>
</div>
<div>
<h3>Solution</h3>
<table data-label="" summary="This figure has two columns, with written instructions on the left and math on the right. At the top of the figure on the right is the equation negative n equals 9. One line down on the left, the instructions say: “Remember negative n is equivalent to negative 1 times n.” To the right of this sentence is the equation negative 1 n equals 9. Another line down to the left, the instructions say: “Divide both sides by negative 1.” To the right of this sentence is the equation divided by negative 1 on both sides: negative 1n over negative 1 equals 9 over negative 1, with divided by negative 1 written in red on both sides. Another line down on the left, the instructions say: “Divide.” To the right of this word is the answer to the equation: n equals negative 9. Below this is the text: “Notice that there are two other ways to solve negative n equals 9. We can also solve this equation by multiplying both sides by negative 1 and also by taking the opposite of both sides.” Another line down on the left, the instructions say: “Check”. To the right of this word is the original equation again: negative n equals 9. Then the instructions say to substitute negative 9 for n, and to the right is the equation with negative 9 substituted in for n: negative negative 9, with negative 9 in parentheses and written in red, might equal 9. One more line down, the instructions say: “Simplify.” To the right of that instruction is the equation 9 equals 9, with a check mark next to it.">
<tbody>
<tr>
<td colspan="2"> </td>
<td><img alt="." src="https://data.ulearngo.com/assets/27f2ed3f-b2de-47d3-bf8e-fddd56666207"/></td>
</tr>
<tr>
<td colspan="2">Remember \(-n\) is equivalent to \(-1n\).</td>
<td><img alt="." src="https://data.ulearngo.com/assets/e67a432c-bbec-4f69-8345-f5e8a59e6b9d"/></td>
</tr>
<tr>
<td colspan="2">Divide both sides by \(-1\).</td>
<td><img alt="." src="https://data.ulearngo.com/assets/02ebe599-3e82-4ece-8804-86338ad94643"/></td>
</tr>
<tr>
<td colspan="2">Divide.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/1482266d-0aef-4343-b14d-b984e6a5f3ad"/></td>
</tr>
<tr>
<td colspan="3">Notice that there are two other ways to solve \(-n=9\). We can also solve this equation by multiplying both sides by \(-1\) and also by taking the opposite of both sides.</td>
</tr>
<tr>
<td>Check:</td>
<td><img alt="." src="https://data.ulearngo.com/assets/f1cadfac-bbb2-4414-bb4e-6aa3939598f8"/></td>
<td> </td>
</tr>
<tr>
<td>Substitute \(n=-9\).</td>
<td><img alt="." src="https://data.ulearngo.com/assets/a24416b0-5fa0-4c4b-bde3-77990dee6a35"/></td>
<td> </td>
</tr>
<tr>
<td>Simplify.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/cc392a29-f657-4a58-ac53-6fe3d586a5dd"/></td>
<td> </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: \(\frac{3}{4}x=12.\)</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 \(\frac{3}{4}\).</p>
<table data-label="" summary="This figure has two columns, with written instructions on the left and math on the right. At the top of the figure on the right is the equation three-fourths x equals 12. One line down on the left, the instructions say: “Multiply by the reciprocal of ¾.” To the right of this sentence is the same equation multiplied by 4/3 on both sides: 4/3 times three-fourths x equals 4/3 times 12, with times 4/3 written in red on both sides. Another line down on the left, the instructions say: “Reciprocals multiply to 1.” To the right of this sentence is the equation 1x equals 4/3 times 12. Another line down on the left, the instructions say: “Multiply.” To the right of this instruction is the answer to the equation: x equals 16. Below this is the text: “Notice that we could have divided both sides of the equation three-fourths x equals 12 by ¾ to isolate x. While this would work, most people would find multiplying by the reciprocal easier.” Two lines down, the instructions say: “Check”. To the right of this word is the original equation again: three-fourths x equals 12. Then the instructions say: “Substitute x equals 16,” and to the right is the equation with 16 substituted in for x: ¾ times 16 (written in red) might equal 12. Below this is the equation 12 equals 12, with a check mark next to it.">
<tbody>
<tr>
<td colspan="2"> </td>
<td><img alt="." src="https://data.ulearngo.com/assets/b2ce96fa-d797-4e85-9169-4ec227cf39de"/></td>
</tr>
<tr>
<td colspan="2">Multiply by the reciprocal of \(\frac{3}{4}\).</td>
<td><img alt="." src="https://data.ulearngo.com/assets/bd41a2b3-7a08-49f0-8ac8-a5dacc26df0b"/></td>
</tr>
<tr>
<td colspan="2">Reciprocals multiply to 1.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/440c5212-2ac0-41d0-9b81-d83d86a23cb6"/></td>
</tr>
<tr>
<td colspan="2">Multiply.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/b16719f9-ecbc-4727-a26c-163e7bf11bb2"/></td>
</tr>
<tr>
<td colspan="3">Notice that we could have divided both sides of the equation \(\frac{3}{4}x=12\) by \(\frac{3}{4}\) to isolate \(x\). While this would work, most people would find multiplying by the reciprocal easier.</td>
</tr>
<tr>
<td>Check:</td>
<td><img alt="." src="https://data.ulearngo.com/assets/93fc017b-082a-44f4-b5f4-8828a5ffd28b"/></td>
<td> </td>
</tr>
<tr>
<td>Substitute \(x=16\).</td>
<td><img alt="." src="https://data.ulearngo.com/assets/806ae555-ca8d-4703-9606-ef8729660a32"/></td>
<td> </td>
</tr>
<tr>
<td> </td>
<td><img alt="." src="https://data.ulearngo.com/assets/2e73c5ce-1012-4b86-b972-e6301c7915ec"/></td>
<td> </td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
</div>
<p>In the next example, all the variable terms are on the right side of the equation. As always, our goal in solving the equation is to isolate the variable.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Solve: \(\frac{8}{15}=-\frac{4}{5}x.\)</p>
</div>
<div>
<h3>Solution</h3>
<table data-label="" summary="This figure has two columns, with written instructions on the left and math on the right. At the top of the figure on the right is the equation 8/15 equals negative four-fifths x. One line down on the left, the instructions say: “Multiply by the reciprocal of negative 4/5.” To the right of this sentence is the same equation with negative 5/4 being multiplied to both sides: negative 5/4 (in parentheses and written in red) times 8/15 equals negative 5/4 (in parentheses and written in red) times negative four-fifths x. Another line down to the left, the instructions say: “Reciprocals multiply to 1.” To the right of this sentence is the equation negative product of 5 times 4 times 2 over the product of 4 times 3 times 5 equals 1x, with the 5s and the 4s canceled out of the numerator and denominator. Another line down to the left, the instructions say: “Multiply.” To the right of this instruction is the equation negative 2/3 equals x. Two lines down, the instructions say: “Check”. To the right of this word is the original equation again: 8/15 equals negative four-fifths x. Then the instructions say: “Let x equal negative 2/3,” and to the right is the equation with negative 2/3 substituted in for x: 8/15 equals negative 4/5 times negative 2/3, with negative 2/3 in parentheses and written in red. Below this is the equation 8/15 equals 8/15, with a check mark next to it.">
<tbody>
<tr>
<td colspan="2"> </td>
<td><img alt="." src="https://data.ulearngo.com/assets/1aee5b6b-2340-4aee-bdc5-e237dea32581"/></td>
</tr>
<tr>
<td colspan="2">Multiply by the reciprocal of \(-\frac{4}{5}\).</td>
<td><img alt="." src="https://data.ulearngo.com/assets/c26c7b95-4b15-4efe-8b1f-641e97742d69"/></td>
</tr>
<tr>
<td colspan="2">Reciprocals multiply to 1.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/4d228b8e-08da-4555-98e3-16702d3da683"/></td>
</tr>
<tr>
<td colspan="2">Multiply.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/d781bf15-d574-4a92-a6b4-cb3fd542850e"/></td>
</tr>
<tr>
<td>Check:</td>
<td><img alt="." src="https://data.ulearngo.com/assets/4ed7798f-5e34-43d0-9d94-9403a226ebea"/></td>
<td> </td>
</tr>
<tr>
<td>Let \(x=-\frac{2}{3}\).</td>
<td><img alt="." src="https://data.ulearngo.com/assets/0f75a116-736c-4185-83ce-fd9ca022ac9f"/></td>
<td> </td>
</tr>
<tr>
<td> </td>
<td><img alt="." src="https://data.ulearngo.com/assets/1800331e-ffa5-4d82-8f98-a76ccd59a629"/></td>
<td> </td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
</div>

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

Explore different ways to solve linear equations, including using addition/subtraction and multiplication/division properties of equality and inequalities, simplifying, translating and solving word problems, and graphing inequalities on a number line.


Solving Equations Using the Division and Multiplication Properties of Equality

Track Your Learning Progress

Earlier discussion