7.3 Distributive Property

Simplify Expressions Using the Distributive Property

Suppose three friends are going to the movies. They each need $$9.25;$ that is, $9$ dollars and $1$ quarter. How much money do they need all together? You can think about the dollars separately from the quarters.

The image shows the equation 3 times 9 equal to 27. Below the 3 is an image of three people. Below the 9 is an image of 9 one dollar bills. Below the 27 is an image of three groups of 9 one dollar bills for a total of 27 one dollar bills.

The image shows the equation 3 times 25 cents equal to 75 cents. Below the 3 is an image of three people. Below the 25 cents is an image of a quarter. Below the 75 cents is an image of three quarters.

They need $3$ times $$9,$ so $$27,$ and $3$ times $1$ quarter, so $75$ cents. In total, they need $$27.75.$

If you think about doing the math in this way, you are using the Distributive Property.

Back to our friends at the movies, we could show the math steps we take to find the total amount of money they need like this:

$\begin{matrix} 3 (9.25) \\ 3 (9 & + & 0.25) \\ 3 (9) & + & 3 (0.25) \\ 27 & + & 0.75 \\ 27.75 \end{matrix}$

In algebra, we use the Distributive Property to remove parentheses as we simplify expressions. For example, if we are asked to simplify the expression $3 open parenthesis x plus 4 close parenthesis,$ the order of operations says to work in the parentheses first. But we cannot add $x$ and $4,$ since they are not like terms. So we use the Distributive Property, as shown in Example 1.

Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in Example 1 would look like this:

The image shows the expression x plus 4 in parentheses with the number 3 outside the parentheses on the left. There are two arrows pointing from the top of the three. One arrow points to the top of the x. The other arrow points to the top of the 4.

The image shows and equation. On the left side of the equation is the expression x plus 4 in parentheses with the number 3 outside the parentheses on the left. There are two arrows pointing from the top of the three. One arrow points to the top of the x. The other arrow points to the top of the 4. This is set equal to 3 times x plus 3 times 4.

The distributive property can be used to simplify expressions that look slightly different from $a (b + c) .$ Here are two other forms.

Do you remember how to multiply a fraction by a whole number? We’ll need to do that in the next two examples.

Using the Distributive Property as shown in the next example will be very useful when we solve money applications later.

In the next example we’ll multiply by a variable. We’ll need to do this in a later chapter.

Example 7

Simplify: $m open parenthesis n minus 4 close parenthesis.$


Distribute.
Multiply.

Notice that we wrote $m times 4 as 4 m.$ We can do this because of the Commutative Property of Multiplication. When a term is the product of a number and a variable, we write the number first.

The next example will use the ‘backwards’ form of the Distributive Property, $(b + c) a = b a + c a .$

When you distribute a negative number, you need to be extra careful to get the signs correct.

In the next example, we will show how to use the Distributive Property to find the opposite of an expression. Remember, $- a = −1 \cdot a .$

Sometimes we need to use the Distributive Property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the distributive property, which removes the parentheses. The next two examples will illustrate this.

Evaluate Expressions Using the Distributive Property

Some students need to be convinced that the Distributive Property always works.

In the examples below, we will practice evaluating some of the expressions from previous examples; in part ⓐ, we will evaluate the form with parentheses, and in part ⓑ we will evaluate the form we got after distributing. If we evaluate both expressions correctly, this will show that they are indeed equal.

Example 14

When $y equals 10$ evaluate: ⓐ $6 open parenthesis 5 y plus 1 close parenthesis$ ⓑ $6 \cdot 5 y + 6 \cdot 1 .$

ⓐ
	$6 open parenthesis 5 y plus 1 close parenthesis$

Simplify in the parentheses.	$6 open parenthesis 51 close parenthesis$
Multiply.	$306$

ⓑ


Simplify.
Add.

Notice, the answers are the same. When $y equals 10,$

$6 (5 y + 1) = 6 \cdot 5 y + 6 \cdot 1 .$

Try it yourself for a different value of $y .$

Example 15

When $y equals 3,$ evaluate ⓐ $−2 open parenthesis 4 y plus 1 close parenthesis$ ⓑ $−2 \cdot 4 y + (−2) \cdot 1 .$

ⓐ
	$−2 open parenthesis 4 y plus 1 close parenthesis$

Simplify in the parentheses.	$−2 open parenthesis 13 close parenthesis$
Multiply.	$−26$

ⓑ
	$−2 \cdot 4 y + (−2) \cdot 1$

Multiply.	$−24 minus 2$
Subtract.	$−26$
The answers are the same. When $y equals 3,$	$−2 (4 y + 1) = −8 y - 2$

Example 16

When $y equals 35$ evaluate ⓐ $− open parenthesis y plus 5 close parenthesis$ and ⓑ $− y minus 5$ to show that $- (y + 5) = − y - 5 .$

ⓐ
	$− open parenthesis y plus 5 close parenthesis$

Add in the parentheses.	$minus open parenthesis 40 close parenthesis$
Simplify.	$−40$

ⓑ
	$- y - 5$

Simplify.	$−40$
The answers are the same when $y equals 35,$ demonstrating that	$- (y + 5) = − y - 5$

Key Concepts

Distributive Property:
- If a,b,c are real numbers then
  - $a (b + c) = a b + a c$
  - $(b + c) a = b a + c a$
  - $a (b - c) = a b - a c$

Practice Makes Perfect

Simplify Expressions Using the Distributive Property

In the following exercises, simplify using the distributive property.

$4 open parenthesis x plus 8 close parenthesis$

$3 open parenthesis a plus 9 close parenthesis$

3a + 27

$8 open parenthesis 4 y plus 9 close parenthesis$

$9 open parenthesis 3 w plus 7 close parenthesis$

27w + 63

$6 open parenthesis c minus 13 close parenthesis$

$7 open parenthesis y minus 13 close parenthesis$

7y − 91

$7 open parenthesis 3 p minus 8 close parenthesis$

$5 open parenthesis 7 u minus 4 close parenthesis$

35u − 20

$\frac{1}{2} (n + 8)$

$\frac{1}{3} (u + 9)$

$\frac{1}{3} u + 3$

$\frac{1}{4} (3 q + 12)$

$\frac{1}{5} (4 m + 20)$

$\frac{4}{5} m + 4$

$9 (\frac{5}{9} y - \frac{1}{3})$

$10 (\frac{3}{10} x - \frac{2}{5})$

3x − 4

$12 (\frac{1}{4} + \frac{2}{3} r)$

$12 (\frac{1}{6} + \frac{3}{4} s)$

2 + 9s

$r open parenthesis s minus 18 close parenthesis$

$u open parenthesis v minus 10 close parenthesis$

uv − 10u

$open parenthesis y plus 4 close parenthesis p$

$open parenthesis a plus 7 close parenthesis x$

ax + 7x

$−2 open parenthesis y plus 13 close parenthesis$

$−3 open parenthesis a plus 11 close parenthesis$

−3a − 33

$−7 open parenthesis 4 p plus 1 close parenthesis$

$−9 open parenthesis 9 a plus 4 close parenthesis$

−81a − 36

$−3 open parenthesis x minus 6 close parenthesis$

$−4 open parenthesis q minus 7 close parenthesis$

−4q + 28

$−9 open parenthesis 3 a minus 7 close parenthesis$

$−6 open parenthesis 7 x minus 8 close parenthesis$

−42x + 48

$- (r + 7)$

$- (q + 11)$

−q − 11

$- (3 x - 7)$

$- (5 p - 4)$

−5p + 4

$5 + 9 (n - 6)$

$12 + 8 (u - 1)$

8u + 4

$16 - 3 (y + 8)$

$18 - 4 (x + 2)$

−4x + 10

$4 - 11 (3 c - 2)$

$9 - 6 (7 n - 5)$

−42n + 39

$22 - (a + 3)$

$8 - (r - 7)$

−r + 15

$−12 - (u + 10)$

$−4 - (c - 10)$

−c + 6

$(5 m - 3) - (m + 7)$

$(4 y - 1) - (y - 2)$

3y + 1

$5 (2 n + 9) + 12 (n - 3)$

$9 (5 u + 8) + 2 (u - 6)$

47u + 60

$9 (8 x - 3) - (−2)$

$4 (6 x - 1) - (−8)$

24x + 4

$14 (c - 1) - 8 (c - 6)$

$11 (n - 7) - 5 (n - 1)$

6n − 72

$6 (7 y + 8) - (30 y - 15)$

$7 (3 n + 9) - (4 n - 13)$

17n + 76

Evaluate Expressions Using the Distributive Property

In the following exercises, evaluate both expressions for the given value.

If $v equals −2,$ evaluate

ⓐ $6 open parenthesis 4 v plus 7 close parenthesis$
ⓑ $6 \cdot 4 v + 6 \cdot 7$

If $u equals −1,$ evaluate

ⓐ $8 open parenthesis 5 u plus 12 close parenthesis$
ⓑ $8 \cdot 5 u + 8 \cdot 12$

ⓐ 56
ⓑ 56

If $n = \frac{2}{3},$ evaluate

ⓐ $3 (n + \frac{5}{6})$
ⓑ $3 \cdot n + 3 \cdot \frac{5}{6}$

If $y = \frac{3}{4},$ evaluate

ⓐ $4 (y + \frac{3}{8})$
ⓑ $4 \cdot y + 4 \cdot \frac{3}{8}$

ⓐ $\frac{9}{2}$
ⓑ $\frac{9}{2}$

If $y = \frac{7}{12},$ evaluate

ⓐ $−3 open parenthesis 4 y plus 15 close parenthesis$
ⓑ $−3 \cdot 4 y + (−3) \cdot 15$

If $p = \frac{23}{30},$ evaluate

ⓐ $−6 open parenthesis 5 p plus 11 close parenthesis$
ⓑ $−6 \cdot 5 p + (−6) \cdot 11$

ⓐ −89
ⓑ −89

If $m equals 0.4,$ evaluate

ⓐ $−10 open parenthesis 3 m minus 0.9 close parenthesis$
ⓑ $−10 \cdot 3 m - (−10) (0.9)$

If $n equals 0.75,$ evaluate

ⓐ $−100 open parenthesis 5 n plus 1.5 close parenthesis$
ⓑ $−100 \cdot 5 n + (−100) (1.5)$

ⓐ −525
ⓑ −525

If $y equals −25,$ evaluate

ⓐ $- (y - 25)$
ⓑ $- y + 25$

If $w equals −80,$ evaluate

ⓐ $- (w - 80)$
ⓑ $- w + 80$

ⓐ 160
ⓑ 160

If $p equals 0.19,$ evaluate

ⓐ $- (p + 0.72)$
ⓑ $- p - 0.72$

If $q equals 0.55,$ evaluate

ⓐ $- (q + 0.48)$
ⓑ $- q - 0.48$

ⓐ −1.03
ⓑ −1.03

Everyday Math

Buying by the case Joe can buy his favorite ice tea at a convenience store for $$1.99$ per bottle. At the grocery store, he can buy a case of $12$ bottles for $$23.88.$

ⓐ Use the distributive property to find the cost of $12$ bottles bought individually at the convenience store. (Hint: notice that $$1.99$ is $$2 minus $0.01.$ )
ⓑ Is it a bargain to buy the iced tea at the grocery store by the case?

Multi-pack purchase Adele’s shampoo sells for $$3.97$ per bottle at the drug store. At the warehouse store, the same shampoo is sold as a $3-pack$ for $$10.49.$

ⓐ Show how you can use the distributive property to find the cost of $3$ bottles bought individually at the drug store.
ⓑ How much would Adele save by buying the $3-pack$ at the warehouse store?

ⓐ 3(4 − 0.03) = 11.91
ⓑ $1.42

Writing Exercises

Simplify $8 (x - \frac{1}{4})$ using the distributive property and explain each step.

Explain how you can multiply $4 open parenthesis $5.97 close parenthesis$ without paper or a calculator by thinking of $$5.97$ as $6 minus 0.03$ and then using the distributive property.

Answers will vary.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

A self-assessment table for students to rate their understanding of simplifying and evaluating expressions using the Distributive Property. — **Figure 7.7**

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

	$3 open parenthesis x plus 4 close parenthesis$
Distribute.	$3 \cdot x + 3 \cdot 4$
Multiply.	$3 x plus 12$


Distribute.	$A mathematical expression shows 8 multiplied by the fraction 3/8 times x, plus 8 multiplied by the fraction 1/4.$
Multiply.


Multiplying by −1 results in the opposite.
Distribute.
Simplify.
Simplify.