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Lesson 3.3 Lesson Tutorials Expressions with the same value, like 12 + 7 and 7 + 12, are equivalent expressions. You can use the Commutative and Associative Properties to write equivalent expressions. Key Vocabulary equivalent expressions, p. 128 Commutative Properties Changing the order of addends or factors does not change the sum or product. Words Numbers 5+8=8+5 ⋅ Algebra ⋅ a+b=b+a ⋅ 5 8=8 5 ⋅ a b=b a Associative Properties Changing the grouping of addends or factors does not change the sum or product. Words Numbers (7 + 4) + 2 = 7 + (4 + 2) ⋅ ⋅ ⋅ ⋅ (7 4) 2 = 7 (4 2) Algebra (a + b) + c = a + (b + c) ⋅ ⋅ ⋅ ⋅ (a b) c = a (b c) EXAMPLE 1 Using Properties to Write Equivalent Expressions a. Simplify the expression 7 + (12 + x). Study Tip 7 + (12 + x) = (7 + 12) + x One way to check whether expressions are equivalent is to evaluate each expression for any value of the variable. In Example 1(a), use x = 2. 7 + (12 + x) = 19 + x = 19 + x 21 = 21 ✓ Add 7 and 12. b. Simplify the expression (6.1 + x) + 8.4. (6.1 + x) + 8.4 = (x + 6.1) + 8.4 ? 7 + (12 + 2) = 19 + 2 Associative Property of Addition Commutative Property of Addition = x + (6.1 + 8.4) Associative Property of Addition = x + 14.5 Add 6.1 and 8.4. c. Simplify the expression 5(11y). ⋅ 5(11y) = (5 11)y Associative Property of Multiplication = 55y Multiply 5 and 11. Simplify the expression. Explain each step. Exercises 5 – 8 128 Chapter 3 ms_green pe_0303.indd 128 1. 10 + (a + 9) 2. ( ) 2 3 1 2 c+— +— 3. 5(4n) Algebraic Expressions and Properties 1/28/15 1:32:08 PM Lesson 3.4 Lesson Tutorials Key Vocabulary like terms, p. 136 Distributive Property Words To multiply a sum or difference by a number, multiply each number in the sum or difference by the number outside the parentheses. Then evaluate. Numbers 3(7 + 2) = 3 × 7 + 3 × 2 a(b + c) = ab + ac Algebra 3(7 − 2) = 3 × 7 − 3 × 2 EXAMPLE 1 a(b − c) = ab − ac Using Mental Math Use the Distributive Property and mental math to find 8 × 53. 8 × 53 = 8(50 + 3) EXAMPLE 2 Write 53 as 50 + 3. = 8(50) + 8(3) Distributive Property = 400 + 24 Multiply. = 424 Add. Using the Distributive Property 1 2 3 4 Use the Distributive Property to find — × 2 —. 1 2 3 4 ( ) ( ) ( ) 1 2 3 4 3 4 1 2 1 2 3 4 Rewrite 2 — as the sum 2 + —. — × 2— = — × 2 + — 3 4 = —×2 + —×— Distributive Property 3 8 =1+— Multiply. 3 8 = 1— Add. Use the Distributive Property to find the product. Exercises 5 –16 1. 5 × 41 2. 9 × 19 3. 6(37) 1 2 5. — × 4— 1 5 6. — × 3— 2 3 4. — × 1— 134 Chapter 3 ms_green pe_0304.indd 134 1 4 2 7 3 4 Algebraic Expressions and Properties 1/28/15 1:33:37 PM