7.3 Distributive Property

The topics covered in this section are:

1. Simplify expressions using the distributive property
2. Evaluate expressions using the distributive property

7.3.1 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.

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.

DISTRIBUTIVE PROPERTY

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

$a(b+c)=ab+ac$

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:

$3(9.25)$

$3(9+0.25)$

$3(9)+3(0.25)$

$27+0.75$

$27.75$

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

Example 1

Simplify: $3(x+4)$.

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:

Example 2

Simplify: $6(5y+1)$.

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

DISTRIBUTIVE PROPERTY

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

$a(b+c)=ab+ac$

Other forms

$a(b-c)=ab-ac$

$(b+c)a=ba+ca$

Example 3

Simplify: $2(x-3)$.

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

Example 4

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

Example 5

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

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

Example 6

Simplify: $100(0.3+0.25q)$.

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

Example 7

Simplify: $m(n-4)$.

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

Example 8

Simplify: $(x+8)p$.

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

Example 9

Simplify: $-2(4y+1)$.

Example 10

Simplify: $-11(4-3a)$.

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

Example 11

Simplify: $-(y+5)$.

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.

Example 12

Simplify: $8-2(x+3)$.

Example 13

Simplify: $4(x-8)-(x+3)$.

7.3.2 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 1., we will evaluate the form with parentheses, and in Part 2. 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 $y=10$ evaluate:

1. $6(5y+1)$
2. $6 \cdot 5y + 6 \cdot 1$

Example 15

When $y=3$, evaluate:

1. $-2(4y+1)$
2. $-2 \cdot 4y + (-2) \cdot 1$

Example 16

When $y=35$ evaluate:

1. $-(y+5)$ and
2. $-y-5$

to show that $-(y+5)=-y-5$.