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Section 6.1 The Fundamentals of Algebra Math in Our World

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Learning Objectives Identify terms and coefficients. Simplify algebraic expressions. Evaluate algebraic expressions. Apply evaluating expressions to real-world situations.

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Algebra A variable, usually represented by a letter, is a quantity that can change. It represents unknown values in a situation. An algebraic expression is a combination of variables, numbers, operation symbols, and grouping symbols. Some examples of algebraic expressions are

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Algebraic Expressions Algebraic expressions are made up of one or more terms. Terms are the pieces in an expression that are separated by addition or subtraction signs. In the expression 8x 2 + 6x - 3, each of 8x 2, 6x, and - 3 is a term. The expression 111 has just one term, namely 111. Every term has a numerical coefficient, or just coefficient. This is the number part of a term, like the 8 in 8x 2.

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EXAMPLE 1 Identifying Terms and Coefficients Identify the terms of the algebraic expression, and the coefficient for each term.

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EXAMPLE 2 The Distributive Property Use the distributive property to multiply out the parentheses. (a) 5(3x + 7)(b) - 3(6A - 7B + 10)

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Simplifying Algebraic Expressions When two terms have the same variables with the same exponents, we will call them like terms. To add or subtract like terms (i.e., combine like terms), add or subtract the numerical coefficients of the like terms.

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EXAMPLE 3 Combining Like Terms Combine like terms for each, if possible. (a)9x – 20x (b)3x 2 + 8x 2 – 2x 2 (c)6x + 8x 2

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EXAMPLE 4 Combining Like Terms Simplify each expression. (a) 9x – 7y + 18 – 27 + 6y – 10x (b) 3x 3 + 4x 2 – 6x + 10 – 7x 2 + 4x 3 + 2x – 6

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EXAMPLE 5 Simplifying an Algebraic Expression Simplify the expression 8(3x 2 + 5) + 3(2 – x) – (5x 2 + x).

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Evaluating Algebraic Expressions Algebraic expressions almost always contain variables, which can be any number. But when we substitute numbers in for the variables, the result is an arithmetic problem. Finding the value of this problem is called evaluating the expression.

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EXAMPLE 6 Evaluating an Expression Evaluate 9x – 3 when x = 5.

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EXAMPLE 7 Evaluating an Expression with Two Variables Evaluate 5x 2 – 7 y + 2 when x = – 3 and y = 6.

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EXAMPLE 8 Finding Commission on Sales A salesperson at a popular clothing store gets a $600 monthly salary and a 10% commission on everything she sells. The expression 0.10x + 600 describes the amount of money she earns each month, where x represents the dollar amount of sales. If she had net sales of $13,240 in July, how much did she earn?

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EXAMPLE 9 Computing Total Cost Including Tax The state of Florida has a sales tax of 6%. (a) Write an expression for the total cost of an item purchased in Florida, including sales tax. The variable should represent the cost of the item before tax. (b) John bought an iPhone at a store in Hollywood, Florida. The price was $349. What was the total cost John paid, including tax?

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EXAMPLE 10 Computing a Discount Price While shopping at her favorite department store, Carmen found a dress she’s been hoping to buy on the 40%-off clearance rack. (a) Write an algebraic expression representing the new price of the dress before tax, then one for the total price including tax. Use 6% as the sales tax rate. (b) If the original price of the dress was $59, find the discounted price, and the total amount that Sally would pay including tax.

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EXAMPLE 11 Using the Formula for Distance The distance in miles an automobile travels is given by the formula d = rt, where r is the rate (or speed) in miles per hour and t is the time in hours. How far will an automobile travel in 6 hours at a rate of 55 miles per hour?

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