
# 7.3: Distributive Property

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Skills to Develop

• Simplify expressions using the distributive property
• Evaluate expressions using the distributive property

be prepared!

Before you get started, take this readiness quiz.

1. Multiply: 3(0.25). If you missed this problem, review Example 5.15
2. Simplify: 10 − (−2)(3). If you missed this problem, review Example 3.51.
3. Combine like terms: 9y + 17 + 3y − 2. If you missed this problem, review Example 2.22.

### 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.

Definition: 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:

$$\begin{split} 3(9&.25) \\ 3(9 &+ 0.25) \\ 3(9) &+ 3(0.25) \\ 27 &+ 0.75 \\ 27&.75 \end{split}$$

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 7.19.

Example 7.17:

Simplify: 3(x + 4).

##### Solution
 Distribute. 3 • x + 3 • 4 Multiply. 3x + 12

Exercise 7.33:

Simplify: 4(x + 2).

Exercise 7.34:

Simplify: 6(x + 7).

Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in Example 7.17 would look like this:

$$3 \cdot x + 3 \cdot 4$$

Example 7.18:

Simplify: 6(5y + 1).

##### Solution

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

Exercise 7.35:

Simplify: 9(3y + 8).

Exercise 7.36:

Simplify: 5(5w + 9).

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

Definition: 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 7.19:

Simplify: 2(x − 3).

##### Solution

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

Exercise 7.37:

Simplify: 7(x − 6).

Exercise 7.38:

Simplify: 8(x − 5).

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

Example 7.20:

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

##### Solution

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

Exercise 7.39:

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

Exercise 7.40:

Simplify: $$\frac{3}{7}$$(u + 21).

Example 7.21:

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

##### Solution

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

Exercise 7.41:

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

Exercise 7.42:

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

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

Example 7.22:

Simplify: 100(0.3 + 0.25q).

##### Solution

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

Exercise 7.43:

Simplify: 100(0.7 + 0.15p).

Exercise 7.44:

Simplify: 100(0.04 + 0.35d).

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

Example 7.23:

Simplify: m(n − 4).

##### Solution

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

Notice that we wrote m • 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.

Exercise 7.45:

Simplify: r(s − 2).

Exercise 7.46:

Simplify: y(z − 8).

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

Example 7.24:

Simplify: (x + 8)p.

##### Solution

 Distribute. $$px + 8p$$

Exercise 7.47:

Simplify: (x + 2)p.

Exercise 7.48:

Simplify: (y + 4)q.

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

Example 7.25:

Simplify: −2(4y + 1).

##### Solution

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

Exercise 7.49:

Simplify: −3(6m + 5).

Exercise 7.50:

Simplify: −6(8n + 11).

Example 7.26:

Simplify: −11(4 − 3a).

##### Solution
 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?

Exercise 7.51:

Simplify: −5(2 − 3a).

Exercise 7.52:

Simplify: −7(8 − 15y).

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

Example 7.27:

Simplify: −(y + 5).

##### Solution
 Multiplying by −1 results in the opposite. $$-1 (y + 5)$$ Distribute. $$-1 \cdot y + (-1) \cdot 5$$ Simplify. $$-y + (-5)$$ Simplify. $$-y -5$$

Exercise 7.53:

Simplify: −(z − 11).

Exercise 7.54:

Simplify: −(x − 4).

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 7.28:

Simplify: 8 − 2(x + 3).

##### Solution
 Distribute. $$8 - 2 \cdot x - 2 \cdot 3$$ Multiply. $$8 - 2x - 6$$ Combine like terms. $$-2x + 2$$

Exercise 7.55:

Simplify: 9 − 3(x + 2).

Exercise 7.56:

Simplify: 7x − 5(x + 4).

Example 7.29:

Simplify: 4(x − 8) − (x + 3).

##### Solution
 Distribute. $$4x - 32 - x - 3$$ Combine like terms. $$3x - 35$$

Exercise 7.57:

Simplify: 6(x − 9) − (x + 12).

Exercise 7.58:

Simplify: 8(x − 1) − (x + 5).

### 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 (a), we will evaluate the form with parentheses, and in part (b) we will evaluate the form we got after distributing. If we evaluate both expressions correctly, this will show that they are indeed equal.

Example 7.30:

When y = 10 evaluate: (a) 6(5y + 1) (b) 6 • 5y + 6 • 1.

##### Solution

(a) 6(5y + 1)

 Substitute $$\textcolor{red}{10}$$ for y. $$6(5 \cdot \textcolor{red}{10} + 1)$$ Simplify in the parentheses. $$6(51)$$ Multiply. $$306$$

(b) 6 • 5y + 6 • 1

 Substitute $$\textcolor{red}{10}$$ for y. $$6 \cdot 5 \cdot \textcolor{red}{10} + 6 \cdot 1$$ Simplify. $$300 + 6$$ Add. $$306$$

Notice, the answers are the same. When y = 10, 6(5y + 1) = 6 • 5y + 6 • 1. Try it yourself for a different value of y.

Exercise 7.59:

Evaluate when w = 3: (a) 5(5w + 9) (b) 5 • 5w + 5 • 9.

Exercise 7.60:

Evaluate when y = 2: (a) 9(3y + 8) (b) 9 • 3y + 9 • 8.

Example 7.31:

When y = 3, evaluate (a) −2(4y + 1) (b) −2 • 4y + (−2) • 1.

##### Solution

(a) −2(4y + 1)

 Substitute $$\textcolor{red}{3}$$ for y. $$-2(4 \cdot \textcolor{red}{3} + 1)$$ Simplify in the parentheses. $$-2(13)$$ Multiply. $$-26$$

(b) −2 • 4y + (−2) • 1

 Substitute $$\textcolor{red}{3}$$ for y. $$-2 \cdot 4 \cdot \textcolor{red}{3} + (-2) \cdot 1$$ Multiply. $$-24 - 2$$ Subtract. $$-26$$ The answers are the same when y = 3. $$-2(4y + 1) = -8y - 2$$

Exercise 7.61:

Evaluate when n = −2: (a) −6(8n + 11) (b) −6 • 8n + (−6) • 11.

Exercise 7.62:

Evaluate when m = −1: (a) −3(6m + 5) (b) −3 • 6m + (−3) • 5.

Example 7.32:

When y = 35 evaluate (a) −(y + 5) and (b) −y − 5 to show that −(y + 5) = −y − 5.

##### Solution

(a) −(y + 5)

 Substitute $$\textcolor{red}{35}$$ for y. $$-(\textcolor{red}{35} + 5)$$ Add in the parentheses. $$-(40)$$ Simplify. $$-40$$

(b) −y − 5

 Substitute $$\textcolor{red}{35}$$ for y. $$-\textcolor{red}{35} - 5$$ Simplify. $$-40$$ The answers are the same when y = 35, demonstrating that $$-(y + 5) = -y - 5$$

Exercise 7.63:

Evaluate when x = 36: (a) −(x − 4) (b) −x + 4 to show that −(x − 4) = − x − 4.

Exercise 7.64:

Evaluate when z = 55: (a) −(z − 10) (b) −z + 10 to show that −(z − 10) = − z + 10.

### Practice Makes Perfect

#### Simplify Expressions Using the Distributive Property

In the following exercises, simplify using the distributive property.

1. 4(x + 8)
2. 3(a + 9)
3. 8(4y + 9)
4. 9(3w + 7)
5. 6(c − 13)
6. 7(y − 13)
7. 7(3p − 8)
8. 5(7u − 4)
9. $$\frac{1}{2}$$(n + 8)
10. $$\frac{1}{3}$$(u + 9)
11. $$\frac{1}{4}$$(3q + 12)
12. $$\frac{1}{5}$$(4m + 20)
13. $$9 \left(\dfrac{5}{9} y − \dfrac{1}{3}\right)$$
14. $$10 \left(\dfrac{3}{10} x − \dfrac{2}{5}\right)$$
15. $$12 \left(\dfrac{1}{4} + \dfrac{2}{3} r\right)$$
16. $$12 \left(\dfrac{1}{6} + \dfrac{3}{4} s\right)$$
17. r(s − 18)
18. u(v − 10)
19. (y + 4)p
20. (a + 7)x
21. −2(y + 13)
22. −3(a + 11)
23. −7(4p + 1)
24. −9(9a + 4)
25. −3(x − 6)
26. −4(q − 7)
27. −9(3a − 7)
28. −6(7x − 8)
29. −(r + 7)
30. −(q + 11)
31. −(3x − 7)
32. −(5p − 4)
33. 5 + 9(n − 6)
34. 12 + 8(u − 1)
35. 16 − 3(y + 8)
36. 18 − 4(x + 2)
37. 4 − 11(3c − 2)
38. 9 − 6(7n − 5)
39. 22 − (a + 3)
40. 8 − (r − 7)
41. −12 − (u + 10)
42. −4 − (c − 10)
43. (5m − 3) − (m + 7)
44. (4y − 1) − (y − 2)
45. 5(2n + 9) + 12(n − 3)
46. 9(5u + 8) + 2(u − 6)
47. 9(8x − 3) − (−2)
48. 4(6x − 1) − (−8)
49. 14(c − 1) − 8(c − 6)
50. 11(n − 7) − 5(n − 1)
51. 6(7y + 8) − (30y − 15)
52. 7(3n + 9) − (4n − 13)

#### Evaluate Expressions Using the Distributive Property

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

1. If v = −2, evaluate
1. 6(4v + 7)
2. 6 · 4v + 6 · 7
2. If u = −1, evaluate
1. 8(5u + 12)
2. 8 · 5u + 8 · 12
3. If n = $$\frac{2}{3}$$, evaluate
1. $$3 \left(n + \dfrac{5}{6}\right)$$
2. 3 • n + 3 • $$\frac{5}{6}$$
4. If y = 3 4 , evaluate
1. 4 ⎛ ⎝ y + 3 8 ⎞ ⎠
2. 4 • y + 4 • $$\frac{3}{8}$$
5. If y = $$\frac{7}{12}$$, evaluate
1. −3(4y + 15)
2. 3 • 4y + (−3) • 15
6. If p = $$\frac{23}{30}$$, evaluate
1. −6(5p + 11)
2. −6 • 5p + (−6) • 11
7. If m = 0.4, evaluate
1. −10(3m − 0.9)
2. −10 • 3m − (−10)(0.9)
8. If n = 0.75, evaluate
1. −100(5n + 1.5)
2. −100 • 5n + (−100)(1.5)
9. If y = −25, evaluate
1. −(y − 25)
2. −y + 25
10. If w = −80, evaluate
1. −(w − 80)
2. −w + 80
11. If p = 0.19, evaluate
1. −(p + 0.72)
2. −p − 0.72
12. If q = 0.55, evaluate
1. −(q + 0.48)
2. −q − 0.48

### Everyday Math

1. 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.

### Self Check

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

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