Prealgebra 2e — Original English

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:

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.

Simplify: 3(x+4).

Solution

Solution

Demonstrates simplifying the expression 3(x+4) using the distributive property.
3(x+4)
Distribute. 3·x+3·4
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.

Simplify: 6(5y+1).

Solution

Solution

A mathematical expression 6(5y + 1) with two blue arrows showing the distribution of 6 to both 5y and 1, illustrating the distributive property.
Distribute. A mathematical expression showing 6 multiplied by 5y, plus 6 multiplied by 1.
Multiply. The image shows the mathematical expression '30y + 6' in black text on a white background. The numbers '30' and '6' are visible, as is the variable 'y' and the plus sign '+'.

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

Simplify: 2(x−3).

Solution

Solution

An image illustrating the distributive property in algebra, showing the expression 2(x - 3) with two blue arrows indicating that 2 should be multiplied by both x and -3.
Distribute. The image shows the mathematical expression 2 multiplied by x minus 2 multiplied by 3.
Multiply. The image shows the mathematical expression '2x - 6' in bold, black text on a white background. The terms '2x' and '6' are separated by a minus sign.

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

Simplify: 34(n+12).

Solution

Solution

Illustration of the distributive property for the expression (3/4)(n + 12), with light blue arrows showing 3/4 being multiplied by 'n' and by '12'.
Distribute. The image shows the mathematical expression three-fourths times n plus three-fourths times twelve, written as (3/4) * n + (3/4) * 12.
Simplify. A mathematical expression displays three-fourths multiplied by 'n', with the result then added to nine, set against a plain white background.

Simplify: 8(38x+14).

Solution

Solution

An image illustrating the distributive property, showing how to multiply 8 by each term inside the parentheses (3/8x and 1/4) in the expression 8(3/8x + 1/4).
Distribute. A mathematical expression shows 8 multiplied by the fraction 3/8 times x, plus 8 multiplied by the fraction 1/4.
Multiply. The image displays the algebraic expression '3x + 2' in black characters against a white background.

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

Simplify: 100(0.3+0.25q).

Solution

Solution

Illustration of the distributive property with the expression 100(0.3 + 0.25q). Arrows show 100 multiplying 0.3 and 0.25q.
Distribute. An algebraic expression is shown, which reads as one hundred multiplied by zero point three, plus one hundred multiplied by zero point two five q.
Multiply. The image displays the mathematical expression '30 + 25q' in a clear, black font against a white background.

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

Simplify: m(n−4).

Solution

Solution

The image shows the algebraic expression m(n-4) with two blue arrows demonstrating the distributive property by indicating that 'm' multiplies both 'n' and '-4' within the parentheses.
Distribute. The mathematical expression m multiplied by n, minus m multiplied by 4, is displayed in bold black font on a white background.
Multiply. The image shows the mathematical expression 'mn - 4m' in a bold, italicized font on a white background.

Notice that we wrote m·4as4m. 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.

Simplify: (x+8)p.

Solution

Solution

An algebraic expression (x + 8)p with arrows illustrating the distributive property, showing 'p' multiplying both 'x' and '8' within the parentheses.
Distribute. The mathematical expression 'px + 8p' is displayed in bold black text on a white background.

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

Simplify: −2(4y+1).

Solution

Solution

The image displays the algebraic expression -2(4y + 1) with blue arrows illustrating the distributive property. The arrows indicate that -2 should be multiplied by both 4y and 1 inside the parentheses.
Distribute. A mathematical expression shows the sum of two products: -2 multiplied by 4y, and -2 multiplied by 1. It is written as -2 4y + (-2) 1.
Simplify. The mathematical expression '-8y - 2' is displayed in a dark gray font against a white background.

Simplify: −11(4−3a).

Solution

Solution

A mathematical expression shows -11 multiplied by the quantity (4 - 3a).
Distribute. A mathematical expression showing the subtraction of two terms, where the first term is -11 multiplied by 4, and the second term is -11 multiplied by 3a. The expression is -11 * 4 - (-11) * 3a.
Multiply. The image displays the mathematical expression '-44 - (-33a)' on a white background, demonstrating subtraction involving a negative number and a variable term enclosed in parentheses.
Simplify. The mathematical expression -44 + 33a is shown on a white background.

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·a.

Simplify: −(y+5).

Solution

Solution

A mathematical expression shows the negative of the sum of 'y' and '5', written as -(y+5).
Multiplying by −1 results in the opposite. The mathematical expression -1(y + 5) is displayed in black text on a white background, representing a negative one multiplied by the sum of y and five.
Distribute. A mathematical expression showing the term -1 multiplied by 'y', added to the product of -1 and 5. The full expression is -1*y + (-1)*5.
Simplify. The image shows the mathematical expression '-y + (-5)', which represents the addition of negative y and negative 5.
Simplify. The mathematical expression -y - 5 is displayed in black characters on a white background.

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.

Simplify: 8−2(x+3).

Solution

Solution

The image shows the mathematical expression 8 - 2(x + 3).
Distribute. A mathematical expression reads 8 minus 2 multiplied by x, minus 2 multiplied by 3.
Multiply. The image shows the mathematical expression 8 - 2x - 6 in a clear, dark font against a white background.
Combine like terms. The mathematical expression -2x + 2 is shown on a white background. The text is rendered in a clear, bold black font, typical of mathematical notation.

Simplify: 4(x−8)−(x+3).

Solution

Solution

A mathematical expression featuring the terms 4(x - 8) - (x + 3) is displayed on a white background.
Distribute. The image shows a mathematical expression: 4x - 32 - x - 3. It displays a series of numbers and variables connected by subtraction signs, all in a black font against a white background.
Combine like terms. The image shows the mathematical expression '3x - 35' in black text on a white background.

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.

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

Solution

Solution

ⓐ
6(5y+1)
The text reads 'Substitute 10 for y.', with '10' highlighted in red. A mathematical expression 6(5 • 10 + 1) is displayed in black font, with the number 10 highlighted in red, on a white background. It represents a calculation following the order of operations.
Simplify in the parentheses. 6(51)
Multiply. 306
ⓑ
A mathematical expression showing the distributive property: 6 multiplied by 5y, plus 6 multiplied by 1. The expression is 6 * 5y + 6 * 1, rendered in a black font on a white background.
The image displays the instruction 'Substitute 10 for y.' written in a blue-green font, with the number '10' highlighted in red, all against a plain white background. A mathematical expression reads '6 times 5 times 10 plus 6 times 1', with the number 10 highlighted in red.
Simplify. The image shows the mathematical expression '300 + 6' in black text against a white background.
Add. The number '306' is displayed in black text on a plain white background.

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.

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

Solution

Solution

ⓐ
−2(4y+1)
The image displays the instruction 'Substitute 3 for y.' in a sans-serif font, with the number 3 highlighted in red, on a plain white background. A mathematical expression is displayed, showing a negative two multiplied by the sum of four times three and one, enclosed in parentheses: -2(4 * 3 + 1). The number '3' is highlighted in red.
Simplify in the parentheses. −2(13)
Multiply. −26
ⓑ
−2·4y+(−2)·1
The text reads 'Substitute 3 for y.' in a sans-serif font, with the number '3' highlighted in red and the rest of the text in a dark teal color. The text is centered on a white background. A mathematical expression shows the calculation '-2 multiplied by 4 multiplied by 3, plus -2 multiplied by 1', with the number 3 highlighted in red.
Multiply. −24−2
Subtract. −26
The answers are the same. When y=3, −2(4y+1)=−8y−2

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

Solution

Solution

ⓐ
−(y+5)
The image displays the text The image displays the mathematical expression -(35 + 5), where 35 is highlighted in red, signifying that 35 and 5 are enclosed within parentheses and preceded by a negative sign.
Add in the parentheses. −(40)
Simplify. −40
ⓑ
−y−5
The text reads 'Substitute 35 for y.' with the number 35 highlighted in red. A mathematical expression showing '-35 - -5' with '-35' in red and '- -5' in black, illustrating subtraction of a negative number.
Simplify. −40
The answers are the same when y=35, demonstrating that −(y+5)=−y−5

Key Concepts

  • Distributive Property:
    • If a,b,c are real numbers then
      • a(b+c)=ab+ac
      • (b+c)a=ba+ca
      • a(b-c)=ab-ac

Practice Makes Perfect

Simplify Expressions Using the Distributive Property

In the following exercises, simplify using the distributive property.

4(x+8)

3(a+9)

Solution

3a + 27

8(4y+9)

9(3w+7)

Solution

27w + 63

6(c−13)

7(y−13)

Solution

7y − 91

7(3p−8)

5(7u−4)

Solution

35u − 20

12(n+8)

13(u+9)

Solution

13u+3

14(3q+12)

15(4m+20)

Solution

45m+4

9(59y−13)

10(310x−25)

Solution

3x − 4

12(14+23r)

12(16+34s)

Solution

2 + 9s

r(s−18)

u(v−10)

Solution

uv − 10u

(y+4)p

(a+7)x

Solution

ax + 7x

−2(y+13)

−3(a+11)

Solution

−3a − 33

−7(4p+1)

−9(9a+4)

Solution

−81a − 36

−3(x−6)

−4(q−7)

Solution

−4q + 28

−9(3a−7)

−6(7x−8)

Solution

−42x + 48

−(r+7)

−(q+11)

Solution

−q − 11

−(3x−7)

−(5p−4)

Solution

−5p + 4

5+9(n−6)

12+8(u−1)

Solution

8u + 4

16−3(y+8)

18−4(x+2)

Solution

−4x + 10

4−11(3c−2)

9−6(7n−5)

Solution

−42n + 39

22−(a+3)

8−(r−7)

Solution

−r + 15

−12−(u+10)

−4−(c−10)

Solution

−c + 6

(5m−3)−(m+7)

(4y−1)−(y−2)

Solution

3y + 1

5(2n+9)+12(n−3)

9(5u+8)+2(u−6)

Solution

47u + 60

9(8x−3)−(−2)

4(6x−1)−(−8)

Solution

24x + 4

14(c−1)−8(c−6)

11(n−7)−5(n−1)

Solution

6n − 72

6(7y+8)−(30y−15)

7(3n+9)−(4n−13)

Solution

17n + 76

Evaluate Expressions Using the Distributive Property

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

If v=−2, evaluate
  1. ⓐ 6(4v+7)
  2. ⓑ 6·4v+6·7
If u=−1, evaluate
  1. ⓐ 8(5u+12)
  2. ⓑ 8·5u+8·12
Solution
  1. ⓐ 56
  2. ⓑ 56
If n=23, evaluate
  1. ⓐ 3(n+56)
  2. ⓑ 3·n+3·56
If y=34, evaluate
  1. ⓐ 4(y+38)
  2. ⓑ 4·y+4·38
Solution
  1. ⓐ 92
  2. ⓑ 92
If y=712, evaluate
  1. ⓐ −3(4y+15)
  2. ⓑ −3·4y+(−3)·15
If p=2330, evaluate
  1. ⓐ −6(5p+11)
  2. ⓑ −6·5p+(−6)·11
Solution
  1. ⓐ −89
  2. ⓑ −89
If m=0.4, evaluate
  1. ⓐ −10(3m−0.9)
  2. ⓑ −10·3m−(−10)(0.9)
If n=0.75, evaluate
  1. ⓐ −100(5n+1.5)
  2. ⓑ −100·5n+(−100)(1.5)
Solution
  1. ⓐ −525
  2. ⓑ −525
If y=−25, evaluate
  1. ⓐ −(y−25)
  2. ⓑ −y+25
If w=−80, evaluate
  1. ⓐ −(w−80)
  2. ⓑ −w+80
Solution
  1. ⓐ 160
  2. ⓑ 160
If p=0.19, evaluate
  1. ⓐ −(p+0.72)
  2. ⓑ −p−0.72
If q=0.55, evaluate
  1. ⓐ −(q+0.48)
  2. ⓑ −q−0.48
Solution
  1. ⓐ −1.03
  2. ⓑ −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.

  1. ⓐ Use the distributive property to find the cost of 12 bottles bought individually at the convenience store. (Hint: notice that $1.99 is $2−$0.01.)

  2. ⓑ 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.

  1. ⓐ Show how you can use the distributive property to find the cost of 3 bottles bought individually at the drug store.

  2. ⓑ How much would Adele save by buying the 3-pack at the warehouse store?

Solution
  1. ⓐ 3(4 − 0.03) = 11.91
  2. ⓑ $1.42

Writing Exercises

Simplify 8(x−14) using the distributive property and explain each step.

Explain how you can multiply 4($5.97) without paper or a calculator by thinking of $5.97 as 6−0.03 and then using the distributive property.

Solution

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.

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