296 CHAPTER 6 Algebra, Graphs, and Functions Like Terms Unlike Terms x x 2 , 7 (same variable, x ) x2 , 9 (only first term has a variable) − y8 , y3 (same variable, y ) x y 5 , 6 (different variables) 4, − 10 (both constants) x, 8 (only first term has a variable) − x x 5 , 6 2 2 (same variable with same exponent) x x 2 , 3 3 2 (different exponents) To simplify an expression means to combine like terms by using the commutative, associative, and distributive properties discussed in Chapter 5. For convenience, we list these properties below. Properties of the Real Numbers + = + a b c ab ac ( ) Distributive property + = + a b b a Commutative property of addition = ab ba Commutative property of multiplication + + = + + a b c a b c ( ) ( ) Associative property of addition = ab c a bc ( ) ( ) Associative property of multiplication In Example 1(c) and 1(d), we can rearrange the terms of the expressions by using the commutative and associative properties that were discussed in Section 5.5. In Example 1(c), we generally list the terms in alphabetical order with the constant term, the term without a variable, last. In Example 1(d), we list the terms in descending order of the variable x. This means the exponents on the variable x get lower as the terms go from left to right with the constant term last. Example 1 Combining Like Terms Combine like terms in each expression. a) + x x 3 6 b) − y y 5 2 c) + − + − − x y x y 3 7 3 5 1 d) − + − + − x x x x 2 3 7 5 4 9 2 2 Solution a) We use the distributive property (in reverse) to combine like terms. x x x x 3 6 (3 6) 9 + = + = Distributive property b) y y y y 5 2 (5 2) 3 − = − = c) + − + − − x y x y 3 7 3 5 1 = + + − − − = − − x x y y x y 3 3 5 7 1 4 2 8 Rearrange terms; place like terms together. Combine like terms. d) − + − + − x x x x 2 3 7 5 4 9 2 2 = − − + + − = − + − x x x x x x 2 5 3 4 7 9 3 2 2 2 2 Rearrange terms; place like terms together. Combine like terms. 7 Now try Exercise 19 Did You Know? Broken Bones Imagine yourself walking down a street in Spain during the Middle Ages. You see a sign over a door: “ Algebrista y Sangrador. ” Inside, you would find a person more ready to give you a haircut than help you with your algebra. The sign translates into “Bonesetter and Bloodletter,” relatively simple medical treatments administered in barbershops of the day. The root word al-jabr , which the North African Muslims brought to Spain along with some concepts of algebra, suggests the restoring of broken parts. The parts might be bones, or they might be mathematical expressions that are broken into separate parts and the parts moved from one side of an equation to the other and reunited in such a way as to make a solution more obvious. Order of Operations To evaluate an expression means to determine the value of the expression for a given value(s) of the variable(s). To evaluate an expression we need to know the order of operations to follow. In Section 5.2 we introduced the order of operations using only numbers. Now we will use the order of operations with expressions that also contain variables.
RkJQdWJsaXNoZXIy NjM5ODQ=