# 7.3 Distributive property

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By the end of this section, you will be able to:
• Simplify expressions using the distributive property
• Evaluate expressions using the distributive property

Before you get started, take this readiness quiz.

1. Multiply: $3\left(0.25\right).$
If you missed this problem, review Decimal Operations
2. Simplify: $10-\left(-2\right)\left(3\right).$
If you missed this problem, review Subtract Integers
3. Combine like terms: $9y+17+3y-2.$
If you missed this problem, review Evaluate, Simplify and Translate Expressions .

## Simplify expressions using the distributive property

Suppose three friends are going to the movies. They each need $\text{9.25};$ 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.

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

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

## Distributive property

If $a,b,c$ are real numbers, then

$a\left(b+c\right)=ab+ac$

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:

$\begin{array}{ccc}\hfill 3\left(9.25\right)& & \\ \hfill 3\left(9& +& 0.25\right)\hfill \\ \hfill 3\left(9\right)& +& 3\left(0.25\right)\hfill \\ \hfill 27& +& 0.75\hfill \\ \hfill 27.75& & \end{array}$

In algebra, we use the Distributive Property to remove parentheses as we simplify expressions. For example, if we are asked to simplify the expression $3\left(x+4\right),$ 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 [link] .

Simplify: $3\left(x+4\right).$

## Solution

 $3\left(x+4\right)$ Distribute. $3·x+3·4$ Multiply. $3x+12$

Simplify: $4\left(x+2\right).$

4 x + 8

Simplify: $6\left(x+7\right).$

6 x + 42

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

Simplify: $6\left(5y+1\right).$

## Solution

 Distribute. Multiply.

Simplify: $9\left(3y+8\right).$

27 y + 72

Simplify: $5\left(5w+9\right).$

25 w + 45

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

## Distributive property

 If $a,b,c$ are real numbers, then $a\left(b+c\right)=ab+ac$ Other forms: $a\left(b-c\right)=ab-ac$ $\left(b+c\right)a=ba+ca$

Simplify: $2\left(x-3\right).$

## Solution

 Distribute. Multiply.

Simplify: $7\left(x-6\right).$

7 x − 42

Simplify: $8\left(x-5\right).$

8 x − 40

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

Simplify: $\frac{3}{4}\left(n+12\right).$

## Solution

 Distribute. Simplify.

Simplify: $\frac{2}{5}\left(p+10\right).$

$\frac{2}{5}p+4$

Simplify: $\frac{3}{7}\left(u+21\right).$

$\frac{3}{7}u+9$

Simplify: $8\left(\frac{3}{8}x+\frac{1}{4}\right).$

## Solution

 Distribute. Multiply.

Simplify: $6\left(\frac{5}{6}y+\frac{1}{2}\right).$

5 y + 3

Simplify: $12\left(\frac{1}{3}n+\frac{3}{4}\right).$

4 n + 9

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

Simplify: $100\left(0.3+0.25q\right).$

## Solution

 Distribute. Multiply.

Simplify: $100\left(0.7+0.15p\right).$

70 + 15 p

Simplify: $100\left(0.04+0.35d\right).$

4 + 35 d

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

Simplify: $m\left(n-4\right).$

## Solution

 Distribute. Multiply.

Notice that we wrote $m·4\phantom{\rule{0.2em}{0ex}}\text{as}\phantom{\rule{0.2em}{0ex}}4m.$ 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.

Simplify: $r\left(s-2\right).$

rs − 2 r

Simplify: $y\left(z-8\right).$

yz − 8 y

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

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