Distributive Property

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.

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.

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: $3(x+4)$.

Solution

$3(x+4)$
Distribute.$3 \cdot x + 3 \cdot 4$
Multiply.$3x+12$

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.

Example 2

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

Solution

.
Distribute.$6 \cdot 5y + 6 \cdot 1$
Multiply.$30y+6$

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: $2(x-3)$.

Solution

.
Distribute.$2 \cdot x – 2 \cdot 3$
Multiply.$2x-6$

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

Solution

.
Distribute.$\frac{3}{4} \cdot n + \frac{3}{4} \cdot 12$
Simplify.$\frac{3}{4} n + 9$

Example 5

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

Solution

.
Distribute.$8 \cdot \frac{3}{8} x + 8 \cdot \frac{1}{4}$
Multiply.$3x+2$

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

Example 6

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

Solution

.
Distribute.$100(0.3)+100(0.25q)$
Multiply.$30+25q$

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

Example 7

Simplify: $m(n-4)$.

Solution

.
Distribute.$m \cdot n – m \cdot 4$
Multiply.$mn-4m$

Notice that we wrote $m \cdot 4$ as $4m$. 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.

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

Example 8

Simplify: $(x+8)p$.

Solution

.
Distribute.$px+8p$

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

Example 9

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

Solution

.
Distribute.$-2 \cdot 4y + (-2) \cdot 1$
Simplify.$-8y-2$

Example 10

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

Solution

$-11(4-3a)$
Distribute.$-11 \cdot 4 – (-11) \cdot 3a$
Multiply.$-44+(33a)$
Simplify.$-44+33a$

You could also write the result as $33a-44$. Do you know why?

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: $-(y+5)$.

Solution

$-(y+5)$
Multiplying by $-1$ results in the opposite.$-1(y+5)$
Distribute.$-1 \cdot y + (-1) \cdot 5$
Simplify.$-y+(-5)$
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: $8-2(x+3)$.

Solution

$8-2(x+3)$
Distribute.$8-2 \cdot x – 2 \cdot 3$
Multiply.$8-2x-6$
Combine like terms.$-2x+2$

Example 13

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

Solution

$4(x-8)-(x+3)$
Distribute.$4x-32-x-3$
Combine like terms.$3x-35$

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. $6 \cdot 5y + 6 \cdot 1$
Solution

Part 1.
$6(5y+1)$
Substitute $10$ for $y$.$6(5 \cdot 10 + 1)$
Simplify in the parentheses.$6(51)$
Multiply.$306$
Part 2.
$6 \cdot 5y + 6 \cdot 1$
Substitute $10$ for $y$.$6 \cdot 5 \cdot 10 + 6 \cdot 1$
Simplify.$300+6$
Add.$306$

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

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

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

Example 15

When $y=3$, evaluate:

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

Part 1.
$-2(4y+1)$
Substitute $3$ for $y$.$-2(4 \cdot 3 + 1)$
Simplify in the parentheses.$-2(13)$
Multiply.$-26$
Part 2.
$-2 \cdot 4y + (-2) \cdot 1$
Substitute $3$ for $y$.$-2 \cdot 4 \cdot 3 + (-2) \cdot 1$
Multiply.$-24-2$
Subtract.$-26$
The answers are the same. When $y=3$.$-2(4y+1)=-8y-2$

Example 16

When $y=35$ evaluate:

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

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

Solution

Part 1.
$-(y+5)$
Substitute $35$ for $y$.$-(35+5)$
Add in the parentheses.$-(40)$
Simplify$-40$
Part 2.
$–y-5$
Substitute $35$ for $y$.$-35-5$
Simplify.$-40$
The answers are the same when $y=35$, demonstrating that$-(y+5)=-y-5$
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