Algebra 1 2 5 Distributive Property Equivalent expressions

Algebra 1 2. 5 Distributive Property

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 Vocabulary

EXAMPLE 1 Apply the distributive property Use the distributive property to write an equivalent expression. 1. 4(y + 3) = 4 y + 12 2. (y + 7)y = y 2 + 7 y 3. n(n – 9) = n 2 – 9 n 4. (2 – n)8 = 16 – 8 n

EXAMPLE 2 Distribute a negative number Use the distributive property to write an equivalent expression. 1. – 2(x + 7)= – 2(x) + – 2(7) = – 2 x – 14 2. (5 – y)(– 3 y) = 5(– 3 y) – y(– 3 y) = – 15 y + 3 y 2 Distribute – 2. Simplify. Distribute – 3 y. Simplify.

EXAMPLE 2 Distribute a negative number 3. –(2 x – 11) = (– 1)(2 x) – (– 1)(11) Multiplicative property of 21 Distribute – 1. = – 2 x + 11 Simplify.

EXAMPLE 3 Identify parts of an expression Identify the terms, like terms, coefficients, and constant terms of the expression 3 x – 4 – 6 x + 2. SOLUTION Terms: 3 x, – 4, – 6 x, 2 Coefficients: 3, – 6 Like terms: 3 x and – 6 x; – 4 and 2 Constant terms: – 4, 2

GUIDED PRACTICE Use the distributive property to write an equivalent expression. 1. 2(x + 3) = 2 x + 6 2. – (4 – y) = – 4 + y 3. (m – 5)(– 3 m) = m (– 3 m) – 5 (– 3 m) Distributive – 3 m = – 3 m 2 + 15 m 1 4. (2 n + 6) 1 = 2 n 1 + 6 2 2 2 =n+3 Distributive – 1 Simplify. Distribute Simplify. 1 2

GUIDED PRACTICE Identify the terms, like terms, coefficients, and constant terms of the expression – 7 y + 8 – 6 y – 13. SOLUTION Terms: – 7 y, 8, – 6 y, – 13 Like terms: – 7 y and – 6 y , 8 and – 13; Coefficients: – 7, – 6 Constant terms: 8, – 13

EXAMPLE 4 Standardized Test Practice Simplify the expression 4(n + 9) – 3(2 + n). A 5 n + 30 B n + 30 C 5 n + 3 4(n + 9) – 3(2 + n) = 4 n + 36 – 3 n = n + 30 D n+3 Distributive property Combine like terms. ANSWER The correct answer is B. A B C D

GUIDED PRACTICE 1. Simplify the expression 5(6 + n) – 2(n – 2). SOLUTION 5(6 + n) – 2(n – 2) = 30 + 5 n – 2 n + 4 = 3 n + 34 Distributive property Combine like terms.

EXAMPLE 5 Solve a multi-step problem 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.

EXAMPLE 5 Solve a multi-step problem STEP 1 Write a verbal model. Then write an equation. Amount burned (calories) C = Burning rate when running (calories/minute) • = 15 C = 15 r + 9(50 – r) Running time (minutes) r + + Burning rate when swimming (calories/minute) • 9 Write equation. = 15 r + 450 – 9 r Distributive property = 6 r + 450 Combine like terms. Swimming time (minutes) (50 – r)

EXAMPLE 5 Solve a multi-step problem STEP 2 Find the value of C when r = 20. Write equation. C = 6 r + 450 = 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.

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 Write a verbal model. Then write an equation. Amount burned (calories) C = = Burning rate when running (calories/minute) • Running time (minutes) 15 C = 15 r + 9 (45 – r) r + + Burning rate when swimming (calories/minute) • 9 Write equation. = 15 r + 405 – 9 r Distributive property = 6 r + 405 Combine like terms. Swimming time (minutes) (45 – r)

GUIDED PRACTICE STEP 2 Find the value of C when r = 20. C = 6 r + 405 = 6(20) + 405 = 525 Write equation. Substitute 20 for r. Then simplify. STEP 3 Find the value of C when r = 30. C = 6 r + 405 = 6(30) + 405 = 585 Write equation. Substitute 30 for r. Then simplify.

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.