Algebra 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 y 2 + 7y n 2 – 9n 16 – 8n

= – 15y + 3y 2 2. (5 – y)(–3y) = 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 – (4 – y) = – 4 + y Distributive – 1 3. (m – 5)(– 3m) = m (– 3m) –5 (– 3m) Distributive – 3m = – 3m m Simplify. 4. (2n + 6) = n = n 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 Practice EXAMPLE 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 + 3 5n + 30 n + 305n + 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 n – 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. = 15 r – 9r Distributive property = 6r 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 problem EXAMPLE 5 C = Write equation. = 6(20) = 570 Substitute 20 for r. Then simplify. ANSWER You burn 570 calories in your 50 minute workout if you run for 20 minutes. STEP 2 Find 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. = 15 r – 9r Distributive property = 6r 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) = 525 Substitute 20 for r. Then simplify. STEP 2 Find the value of C when r = 20. 6r Write equation. = 6(30) = 585 Substitute 30 for r. Then simplify. STEP 3 Find the value of C when r = 30. 6r C =

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.