Algebra 1 Algebra 1 2.5 Distributive Property

Vocabulary

Equivalent expressions: two expressions that have the same output value for every input value

Distributive Property: multiply the outside number to every number in the parenthesis

Term: the individual parts of an expression

Vocabulary

Coefficient: the number part of a term

Constant Term: a term that has a number part but no variable

Like Terms: terms that have the same variable part

Use the distributive property to write an equivalent expression.

EXAMPLE 1Apply the distributive property

1. 4(y + 3) =

2. (y + 7)y =

4. (2 – n)8 =

3. n(n – 9) =

4y + 12

y2 + 7y

n2 – 9n

16 – 8n

= – 15y + 3y2

2. (5 – y)(–3y) =

Simplify.

Simplify.

Distribute – 3y.

= – 2x – 14

Distribute – 2.

Use the distributive property to write an equivalent expression.

EXAMPLE 2Distribute a negative number

1. –2(x + 7)=– 2(x) + – 2(7)

5(–3y) – y(–3y)

Simplify.

= (– 1)(2x) – (–1)(11)

3. –(2x – 11) =of 21

Multiplicative property

EXAMPLE 2Distribute a negative number

Distribute – 1.

= – 2x + 11

(–1)(2x – 11)

Constant terms: – 4, 2

Coefficients: 3, – 6

Like terms: 3x and – 6x; – 4 and 2

Identify the terms, like terms, coefficients, and constant terms of the expression 3x – 4 – 6x + 2.

SOLUTION

EXAMPLE 3 Identify parts of an expression

Terms: 3x, – 4, – 6x, 2

GUIDED PRACTICE

Use the distributive property to write an equivalent expression.

1. 2(x + 3) = 2x + 6

2. – (4 – y) = – 4 + y Distributive – 1

3. (m – 5)(– 3m) = m (– 3m) –5 (– 3m) Distributive – 3m

= – 3m2 + 15m Simplify.

4. (2n + 6) =12

122n + 61

2

= n + 3

12

Distribute

Simplify.

GUIDED PRACTICE

Identify the terms, like terms, coefficients, and constant terms of the expression – 7y + 8 – 6y – 13.

Coefficients: – 7, – 6

Like terms: – 7y and – 6y , 8 and – 13;

SOLUTION

Terms: – 7y, 8, – 6y, – 13

Constant terms: 8, – 13

Standardized Test PracticeEXAMPLE 4

ANSWER

The correct answer is B. DCBA

Simplify the expression 4(n + 9) – 3(2 + n).

4(n + 9) – 3(2 + n) = Distributive property

= n + 30 Combine like terms.

A B C D n + 35n + 30 n + 30 5n + 3

4n + 36 – 6 – 3n

GUIDED PRACTICE

1. Simplify the expression 5(6 + n) – 2(n – 2).

5(6 + n) – 2(n – 2) = Distributive property

= 3n + 34 Combine like terms.

30 + 5n – 2n + 4

SOLUTION

Solve a multi-step problem

EXAMPLE 5

Your daily workout plan involves a total of 50 minutes of running and swimming. You burn 15 calories per minute when running and 9 calories per minute when swimming. Let r be the number of minutes that you run. Find the number of calories you burn in your 50 minute workout if you run for 20 minutes.

SOLUTION

The workout lasts 50 minutes, and your running time is r minutes. So, your swimming time is (50 – r) minutes.

Solve a multi-step problem

EXAMPLE 5

STEP 1

C = Write equation.

= 15r + 450 – 9r Distributive property

= 6r + 450 Combine like terms.

Write a verbal model. Then write an equation.

15r + 9(50 – r) C = 15 r + 9 (50 – r)

Amount burned

(calories)

Burning rate when running

(calories/minute)

Running time

(minutes)

Swimming time

(minutes)= +•

Burning rate when swimming (calories/minute)

•

Solve a multi-step problemEXAMPLE 5

C = Write equation.

= 6(20) + 450 = 570 Substitute 20 for r. Then simplify.

ANSWER

You burn 570 calories in your 50 minute workout if you run for 20 minutes.

STEP 2Find the value of C when r = 20.

6r + 450

GUIDED PRACTICE

WHAT IF…

Suppose your workout lasts 45 minutes. How many calories do you run for 20 minutes? 30 minutes?

SOLUTION

The workout lasts 45 minutes, and your running time is r minutes. So, your swimming time is (45 – r) minutes.

GUIDED PRACTICE

STEP 1

C = 15 r + 9 (45 – r)

C = Write equation.

= 15r + 405 – 9r Distributive property

= 6r + 405 Combine like terms.

Write a verbal model. Then write an equation.

15 r + 9 (45 – r)

Amount burned

(calories)

Burning rate when running

(calories/minute)

Running time

(minutes)

Swimming time

(minutes)= +•

Burning rate when swimming (calories/minute) •

GUIDED PRACTICE

C = Write equation.

= 6(20) + 405 = 525 Substitute 20 for r. Then simplify.

STEP 2Find the value of C when r = 20.

6r + 405

Write equation.

= 6(30) + 405 = 585 Substitute 30 for r. Then simplify.

STEP 3Find the value of C when r = 30.

6r + 405C =

GUIDED PRACTICE

ANSWER

You burn 525 calories in your 45 minute workout if you run for 20 minutes.

You burn 585 calories in your 45 minute workout if you run for 30 minutes.