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The Commutative, Associative, and Distributive Laws Recall: Algebra is the study of how to rewrite mathematical expressions without changing their value. Two different expressions that have the exact same value are said to be equivalent. Goal: Examine and apply three laws of algebra The Commutative Law For addition: If a and b are variables that stand for any number, then a+b=b+a Translate into English: Examples: The Commutative Law The Associative Law For multiplication: For addition: If a and b are variables that stand for any number, then If a, b and c are variables that stand for any number, then ab=ba a + (b + c) = (a + b) + c Translate into English: Translate into English: Examples: Examples: 1 The Associative Law For multiplication: If a, b and c are variables that stand for any number, then Examples: Use both the commutative and associative laws to find equivalent expressions: 1. (x + 3) + 9 a . (b . c) = (a . b) . c Translate into English: 2. 3(t . 4) Examples: The Distributive Law If a, b and c are variables that stand for any number, then a(b + c) = ab + ac Another name: the Distributive Law of Multiplication over Addition. In an algebraic expression, terms are the numbers, variables, or expressions that are added. Examples: List the terms. y s+y 3x + + 5 2 Examples: In an algebraic expression, factors are numbers, variables, or expressions that are multiplied. Examples: List the factors. xyz 2(7 + y) The process of factoring finds an equivalent expression of factors. Example: We know that 2(7 + y) can be written as 14 + 2y using the distributive law. So the factors of 14 + 2y are 2 and (7 + y). 2 Factoring “undoes” multiplication. Examples: Factor. 1. 3y + 9 2. 5 + 15t + 10 3. xy + y 3