3.2 Solve Equations Using the Division and Multiplication Properties of Equality

Pooja Gupta

3.2 Solve Equations Using the Division and Multiplication Properties of Equality

Learning Objectives

By the end of this section, you will be able to:

Solve equations using the Division and Multiplication Properties of Equality
Solve equations that need to be simplified

Solve Equations Using the Division and Multiplication Properties of Equality

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.

Let’s look at our puzzle again with the envelopes and counters in (Figure 1).

Figure 1. Described in the previous paragraph. — Figure .1

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?

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 $6 \div 2=3$ ).

What equation models the situation shown in (Figure 2)? There are two envelopes, and each contains $x$ counters. Together, the two envelopes must contain a total of 6 counters.

Figure 2. Described in the previous paragraph. — Figure .2

	$2x=6$
If we divide both sides of the equation by 2, as we did with the envelopes and counters,	$\frac{2x}{\textcolor{red}{2}} = \frac{6}{\textcolor{red}{2}}$
we get:	$x=3$

We found that each envelope contains 3 counters. Does this check? We know $2\times3=6$ , so it works! Three counters in each of two envelopes does equal six!

This example leads to the Division Property of Equality.

Division and Multiplication Properties of Equality

Division Property of Equality: For all real numbers $a,b,c$ , and $c\ne 0$ , if $a=b$ , then $\frac{a}{c}=\frac{b}{c}$ .

Multiplication Property of Equality: For all real numbers $a,b,c$ , if $a=b$ , then $ac=bc$ .

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

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.

EXAMPLE 1

Solve: $4x=-28$ .

Solution

We use the Division Property of Equality to divide both sides by $4$ .

	$4x=-28$
Divide both sides by 4 to undo the multiplication.	$\frac{4x}{\textcolor{red}{4}} = \frac{-28}{\textcolor{red}{4}}$
Simplify.	$x=-7$
Check your answer. Let $x=-7$ .	$4x=-28$ $4(\textcolor{red}{-7})\stackrel{?}{=}-28$ $-28=-28 \checkmark$

Since this is a true statement, $x=-7$ is a solution to $4x=-28$ .

TRY IT 1.1

Solve: $3y=-48$ .

Show answer

y = −16

TRY IT 1.2

Solve: $4z=-52$ .

Show answer

z = −13

In the previous example, to ‘undo’ multiplication, we divided. How do you think we ‘undo’ division?

EXAMPLE 2

Solve: $\frac{\phantom{\rule{0.4em}{0ex}}a}{-7}=-42$ .

Solution

Here $a$ is divided by $-7$ . We can multiply both sides by $-7$ to isolate $a$ .

	$\frac{a}{-7} = -42$
Multiply both sides by $-7$ .	$\textcolor{red}{-7}(\frac{a}{-7}) = \textcolor{red}{-7} (-42)$ $\frac{-7a}{-7}=294$
Simplify.	$a=294$
Check your answer. Let $a=294$ .	$\frac{a}{-7} = -42$ $\frac{\textcolor{red}{294}}{-7}\stackrel{?}{=}-42$ $-42=-42 \checkmark$

TRY IT 2.1

Solve: $\frac{\phantom{\rule{0.4em}{0ex}}b}{-6}=-24$ .

Show answer

b = 144

TRY IT 2.2

Solve: $\frac{\phantom{\rule{0.4em}{0ex}}c}{-8}=-16$ .

Show answer

c = 128

EXAMPLE 3

Solve: $-r=2$ .

Solution

Remember $-r$ is equivalent to $-1r$ .

	$-r=2$
Rewrite $-r$ as $-1r$ .	$-1r=2$
Divide both sides by $-1$ .	$\frac{-1r}{\textcolor{red}{-1}} = \frac{2}{\textcolor{red}{-1}}$
	$r=-2$
Check.	$-r=2$
Substitute $r=-2$	$-(\textcolor{red}{-2})\stackrel{?}{=}2$
Simplify.	$2 =2 \checkmark$

We see that there are two other ways to solve $-r=2$ .

We could multiply both sides by $-1$ .

We could take the opposite of both sides.

TRY IT 3.1

Solve: $-k=8$ .

Show answer

k = −8

TRY IT 3.2

Solve: $-g=3$ .

Show answer

g = −3

EXAMPLE 4

Solve: $\frac{2}{3}\phantom{\rule{0.1em}{0ex}}x=18$ .

Solution

Since the product of a number and its reciprocal is $1$ , our strategy will be to isolate $x$ by multiplying by the reciprocal of $\frac{2}{3}$ .

	$\frac{2}{3}x=18$
Multiply by the reciprocal of $\frac{2}{3}$ which is $\frac{3}{2}$	$\textcolor{red}{\frac{3}{2}} \cdot \frac{2}{3} = \textcolor{red}{\frac{3}{2}} \cdot 18$
Reciprocals multiply to one.	$1x= \frac{3}{2} \cdot \frac{18}{1}$
Multiply.	$x=27$
Check your answer. Let $x=27$	$\frac{2}{3}=18$ $\frac{2}{3} \cdot \textcolor{red}{27}\stackrel{?}{=} 18$ $18=18 \checkmark$

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.

TRY IT 4.1

Solve: $\frac{2}{5}\phantom{\rule{0.1em}{0ex}}n=14$ .

Show answer

n = 35

TRY IT 4.2

Solve: $\frac{5}{6}\phantom{\rule{0.1em}{0ex}}y=15$ .

Show answer

y = 18

Solve Equations That Need to be Simplified

Many equations start out more complicated than the ones we’ve just solved. First, we need to simplify both sides of the equation as much as possible

EXAMPLE 5

Solve: $8x+9x-5x=-3+15$ .

Solution

Start by combining like terms to simplify each side.

	$8x+9x-5x=-3+15$
Combine like terms.	$12x=12$
Divide both sides by 12 to isolate x.	$\frac{12x}{\textcolor{red}{12}}=\frac{12}{\textcolor{red}{12}}$
Simplify.	$x=1$
Check your answer. Let $x=1$	$8x+9x-5x=-3+15$ $8 \cdot \textcolor{red}{1} +9 \cdot \textcolor{red}{1} - 5 \cdot \textcolor{red}{1}\stackrel{?}{=} -3+15$ $8+9-5 \stackrel{?}{=}12$ $12=12 \checkmark$

TRY IT 5.1

Solve: $7x+6x-4x=-8+26$ .

Show answer

x = 2

TRY IT 5.2

Solve: $11n-3n-6n=7-17$ .

Show answer

n = −5

EXAMPLE 6

Solve: $11-20=17y-8y-6y$ .

Solution

Simplify each side by combining like terms.

	$11-20=17y-8y-6y$
Simplify each side.	$-9=3y$ *Note
Divide both sides by 3 to isolate y.	$\frac{-9}{\textcolor{red}{3}}=\frac{3y}{\textcolor{red}{3}}$
Simplify.	$-3=y$
Check your answer. Let $y=-3$	$11-20=17y-8y-6y$ $11-20\stackrel{?}{=}17(\textcolor{red}{-3})-8(\textcolor{red}{-3})-6(\textcolor{red}{-3})$ $11-20\stackrel{?}{=}-51+24+18$ $-9=-9 \checkmark$

*Notice that the variable ended up on the right side of the equal sign when we solved the equation. You may prefer to take one more step to write the solution with the variable on the left side of the equal sign.

TRY IT 6.1

Solve: $18-27=15c-9c-3c$ .

Show answer

c = −3

TRY IT 6.2

Solve: $18-22=12x-x-4x$ .

Show answer

$x=-\frac{4}{7}$

EXAMPLE 7

Solve: $-3\left(n-2\right)-6=21$ .

Solution

Remember—always simplify each side first.

	$-3(n-2)-6=21$
Distribute.	$-3n+6-6=21$
Simplify.	$-3n+\cancel{6}-\cancel{6}=21$ $-3n=21$
Divide both sides by -3 to isolate n.	$\frac{-3n}{\textcolor{red}{-3}} =\frac{21}{\textcolor{red}{-3}}$ $n=-7$
Check your answer. Let $n=-7$ .	$-3(n-2)-6=21$ $-3(\textcolor{red}{-7}-2)-6\stackrel{?}{=}21$ $-3(-9)-6\stackrel{?}{=}21$ $27-6\stackrel{?}{=}21$ $21=21 \checkmark$

TRY IT 7.1

Solve: $-4\left(n-2\right)-8=24$ .

Show answer

n = −6

TRY IT 7.2

Solve: $-6\left(n-2\right)-12=30$ .

Show answer

n = −5

Key Concepts

Division and Multiplication Properties of Equality
- Division Property of Equality: For all real numbers a, b, c, and $c\ne 0$ , if $a=b$ , then $\frac{a}{c} = \frac{b}{c}$ .
- Multiplication Property of Equality: For all real numbers a, b, c, if $a=b$ , then $ac=bc$ .

Practice Makes Perfect

Solve Equations Using the Division and Multiplication Properties of Equality

In the following exercises, solve each equation for the variable using the Division Property of Equality and check the solution.

1. $7p=63$	2. $8x=32$
3. $-9x=-27$	4. $-5c=55$
5. $-72=12y$	6. $-90=6y$
7. $-8m=-56$	8. $-16p=-64$
9. $0.75a=11.25$	10. $0.25z=3.25$
11. $4x=0$	12. $-3x=0$

In the following exercises, solve each equation for the variable using the Multiplication Property of Equality and check the solution.

13. $\frac{z}{2}=14$	14. $\frac{x}{4}=15$
15. $\frac{\phantom{\rule{0.4em}{0ex}}c}{-3}=-12$	16. $-20=\frac{\phantom{\rule{0.4em}{0ex}}q}{-5}$
17. $\frac{q}{6}=-8$	18. $\frac{y}{9}=-6$
19. $-4=\frac{p}{-20}$	20. $\frac{m}{-12}=5$
21. $\frac{3}{5}\phantom{\rule{0.1em}{0ex}}r=15$	22. $\frac{2}{3}\phantom{\rule{0.1em}{0ex}}y=18$
23. $24=-\frac{3}{4}\phantom{\rule{0.1em}{0ex}}x$	24. $-\frac{5}{8}\phantom{\rule{0.1em}{0ex}}w=40$
25. $-\frac{1}{3}\phantom{\rule{0.1em}{0ex}}q=-\frac{5}{6}$	26. $-\frac{2}{5}=\frac{1}{10}\phantom{\rule{0.1em}{0ex}}a$

Solve Equations That Need to be Simplified

In the following exercises, solve the equation.

27. $6y-3y+12y=-43+28$	28. $8a+3a-6a=-17+27$
29. $-5m+7m-8m=-6+36$	30. $-9x-9x+2x=50-2$
31. $-18-7=5t-9t-6t$	32. $100-16=4p-10p-p$
33. $\frac{5}{12}\phantom{\rule{0.1em}{0ex}}q+\frac{1}{2}\phantom{\rule{0.1em}{0ex}}q=25-3$	34. $\frac{7}{8}\phantom{\rule{0.1em}{0ex}}n-\frac{3}{4}\phantom{\rule{0.1em}{0ex}}n=9+2$
35. $0.05p-0.01p=2+0.24$	36. $0.25d+0.10d=6-0.75$

Everyday Math

37. Teaching Connie’s kindergarten class has $24$ children. She wants them to get into $4$ equal groups. Find the number of children in each group, $g$ , by solving the equation $4g=24$ .	38. Balloons Ramona bought $18$ balloons for a party. She wants to make $3$ equal bunches. Find the number of balloons in each bunch, $b$ , by solving the equation $3b=18$ .
39. Unit price Nishant paid $\text{\$12.96}$ for a pack of $12$ juice bottles. Find the price of each bottle, $b$ , by solving the equation $12b=12.96$ .	40. Ticket price Daria paid $\text{\$36.25}$ for $5$ children’s tickets at the ice skating rink. Find the price of each ticket, $p$ , by solving the equation $5p=36.25$ .
41. Fabric The drill team used $14$ yards of fabric to make flags for one-third of the members. Find how much fabric, $f$ , they would need to make flags for the whole team by solving the equation $\frac{1}{3}\phantom{\rule{0.1em}{0ex}}f=14$ .	42. Fuel economy Tania’s SUV gets half as many miles per gallon (mpg) as her husband’s hybrid car. The SUV gets $\text{18 mpg}$ . Find the miles per gallons, $m$ , of the hybrid car, by solving the equation $\frac{1}{2}\phantom{\rule{0.1em}{0ex}}m=18$ .

Writing Exercises

43. Emiliano thinks $x=40$ is the solution to the equation $\frac{1}{2}\phantom{\rule{0.1em}{0ex}}x=80$ . Explain why he is wrong.

44. Frida started to solve the equation $-3x=36$ by adding $3$ to both sides. Explain why Frida’s method will result in the correct solution.

Answers

1. 9	3. 3	5. −6
7. 7	9. 15	11. 0
13. 28	15. 36	17. −48
19. 80	21. 25	23. −32
25. 5/2	27. y = −1	29. m = −5
31. $t=\frac{5}{2}$	33. q = 24	35. p = 56
37. 6 children	39. $1.08	41. 42 yards
43. Answer will vary.

Attributions

This chapter has been adapted from “Solve Equations Using the Division and Multiplication Properties of Equality” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

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Solve Equations Using the Division and Multiplication Properties of Equality

Solve Equations That Need to be Simplified

Key Concepts

Practice Makes Perfect

Solve Equations Using the Division and Multiplication Properties of Equality

Solve Equations That Need to be Simplified

Everyday Math

Writing Exercises

Answers

Attributions

License

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