Solving Equations That Contain Fractions Using the Multiplication Property of Equality

Learning Outcomes

Use the multiplication and division properties to solve equations with fractions and division

Solve Equations with Fractions Using the Multiplication and Division Properties of Equality

We will solve equations that require multiplication or division to isolate the variable. First, let's consider the division property of equality again.

The Division Property of Equality

For any numbers

a,b

, and

c

\text{if }a=b,\text{ then }\frac{a}{c}=\frac{b}{c}

. If you divide both sides of an equation by the same quantity, you still have equality.

Let's put this idea in practice with an example. We are looking for the number you multiply by

10

to get

44

, and we can use division to find out.

Example

Solve:

10q=44

Answer: Solution:

		$10q=44$
Divide both sides by $10$ to undo the multiplication.		$\frac{10q}{10}=\frac{44}{10}$
Simplify.		$q=\frac{22}{5}$
Check:
Substitute $q=\frac{22}{5}$ into the original equation.	$10\left(\frac{22}{5}\right)\stackrel{?}{=}44$
Simplify.	$\stackrel{2}{\overline{)10}}\left(\frac{22}{\overline{)5}}\right)\stackrel{?}{=}44$
Multiply.	$44=44$

The solution to the equation was the fraction

\frac{22}{5}

. We leave it as an improper fraction.

Try It

#146141 [ohm_question]146141[/ohm_question]

Now, 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 Multiplication Property of Equality 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.

The Multiplication Property of Equality

For any numbers

a,b

, and

c

\text{if }a=b,\text{ then }ac=bc

. If you multiply both sides of an equation by the same quantity, you still have equality.

Let’s use the Multiplication Property of Equality to solve the equation

\frac{x}{7}=-9

Example

Solve:

\frac{x}{7}=-9

. Solution:

		$\frac{x}{7}=-9$
Use the Multiplication Property of Equality to multiply both sides by $7$ . This will isolate the variable.		$\color{red}{7}\cdot\frac{x}{7}=\color{red}{7}(-9)$
Multiply.		$\frac{7x}{7}=-63$
Simplify.		$x=-63$
Check. Substitute $\color{red}{-63}$ for x in the original equation.	$\frac{\color{red}{-63}}{7}\stackrel{?}{=}-9$
The equation is true.	$-9=-9$

Try It

#146147 [ohm_question]146147[/ohm_question]

Example

Solve:

\frac{p}{-8}=-40

Answer: Solution: Here, $p$ is divided by $-8$ . We must multiply by $-8$ to isolate $p$ .

		$\frac{p}{-8}=-40$
Multiply both sides by $-8$		$\color{red}{-8}(\frac{p}{-8})=\color{red}{-8}(-40)$
Multiply.		$\frac{-8p}{-8}=320$
Simplify.		$p=320$
Check:
Substitute $p=320$ .	$\frac{\color{red}{320}}{-8}\stackrel{?}{=}-40$
The equation is true.	$-40=-40$

Try It

#146336 [ohm_question]146336[/ohm_question]

In the following video we show two more examples of when to use the multiplication and division properties to solve a one-step equation. https://youtu.be/BN7iVWWl2y0

Solve Equations with a Coefficient of $-1$

Look at the equation

-y=15

. Does it look as if

y

is already isolated? But there is a negative sign in front of

y

, so it is not isolated. There are three different ways to isolate the variable in this type of equation. We will show all three ways in the next example.

Example

Solve:

-y=15

Answer: Solution: One way to solve the equation is to rewrite $-y$ as $-1y$ , and then use the Division Property of Equality to isolate $y$ .

	$-y=15$
Rewrite $-y$ as $-1y$ .	$-1y = 15$
Divide both sides by $−1$ .	$\frac{-1y}{\color{red}{-1}}=\frac{15}{\color{red}{-1}}$
Simplify each side.	$y = -15$

Another way to solve this equation is to multiply both sides of the equation by

-1

	$y=15$
Multiply both sides by $−1$ .	$\color{red}{-1}(-y)=\color{red}{-1}(15)$
Simplify each side.	$y=-15$

The third way to solve the equation is to read

-y

as "the opposite of

y\text{."}

What number has

15

as its opposite? The opposite of

15

-15

. So

y=-15

. For all three methods, we isolated

y

is isolated and solved the equation. Check:

	$y=15$
Substitute $y=-15$ .	$-(\color{red}{-15})\stackrel{?}{=}(15)$
Simplify. The equation is true.	$15=15$

Try It

#146153 [ohm_question]146153[/ohm_question]

In the next video we shoe more examples of how to solve an equation with a negative variable. https://youtu.be/FJmNpHeOcpo

Solve Equations with a Fraction Coefficient

When we have an equation with a fraction coefficient we can use the Multiplication Property of Equality to make the coefficient equal to

1

. For example, in the equation:

\frac{3}{4}x=24

The coefficient of

x

\frac{3}{4}

. To solve for

x

, we need its coefficient to be

1

. Since the product of a number and its reciprocal is

1

, our strategy here will be to isolate

x

by multiplying by the reciprocal of

\frac{3}{4}

. We will do this in the next example.

Example

Solve:

\frac{3}{4}x=24

Answer: Solution:

		$\frac{3}{4}x = 24$
Multiply both sides by the reciprocal of the coefficient.		$\color{red}{\frac{4}{3}}\cdot\frac{3}{4}x = \color{red}{\frac{4}{3}}\cdot 24$
Simplify.		$1x=\frac{4}{3}\cdot\frac{24}{1}$
Multiply.		$x=32$
Check:	$\frac{3}{4}x=24$
Substitute $x=32$ .	$\frac{3}{4}\cdot 32\stackrel{?}{=}24$
Rewrite $32$ as a fraction.	$\frac{3}{4}\cdot\frac{32}{1}\stackrel{?}{=}24$
Multiply. The equation is true.	$24=24$

Notice that in the equation

\frac{3}{4}x=24

, we could have divided both sides by

\frac{3}{4}

to get

x

by itself. Dividing is the same as multiplying by the reciprocal, so we would get the same result. But most people agree that multiplying by the reciprocal is easier.

Try It

#146156 [ohm_question]146156[/ohm_question]

Example

Solve:

-\frac{3}{8}w=72

Answer: Solution: The coefficient is a negative fraction. Remember that a number and its reciprocal have the same sign, so the reciprocal of the coefficient must also be negative.

		$\frac{3}{8}w=72$
Multiply both sides by the reciprocal of $-\frac{3}{8}$ .		$\color{red}{-\frac{8}{3}}(-\frac{3}{8}w)=(\color{red}{-\frac{8}{3}})72$
Simplify; reciprocals multiply to one.		$1w=-\frac{8}{3}\cdot\frac{72}{1}$
Multiply.		$w=-192$
Check:	$-\frac{3}{8}w=72$
Let $w=-192$ .	$-\frac{3}{8}(-192)\stackrel{?}{=}72$
Multiply. It checks.	$72=72$

Try It

#146158 [ohm_question]146158[/ohm_question]

In the next video example you will see another example of how to use the reciprocal of a fractional coefficient to solve an equation.

https://youtu.be/Ea5eW8rZxEI

Solving Equations That Contain Fractions Using the Multiplication Property of Equality

Learning Outcomes

Solve Equations with Fractions Using the Multiplication and Division Properties of Equality

The Division Property of Equality

Example

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The Multiplication Property of Equality

Example

Try It

Example

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Solve Equations with a Coefficient of $-1$

Example

Try It

Solve Equations with a Fraction Coefficient

Example

Try It

Example

Try It

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Study Guides > Prealgebra

Solving Equations That Contain Fractions Using the Multiplication Property of Equality

Learning Outcomes

Solve Equations with Fractions Using the Multiplication and Division Properties of Equality

The Division Property of Equality

Example

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The Multiplication Property of Equality

Example

Try It

Example

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Solve Equations with a Coefficient of −1-1−1

Example

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Solve Equations with a Fraction Coefficient

Example

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Example

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Licenses & Attributions

CC licensed content, Original

CC licensed content, Shared previously

CC licensed content, Specific attribution

Solve Equations with a Coefficient of $-1$