Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.2 – Slide 1

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.2 – Slide 2 The Real Number System Chapter 1

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.2 – Slide Variables, Expressions, and Equations

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.2 – Slide 4 Objectives 1.Evaluate algebraic expressions, given values for the variables. 2.Translate phrases from words to algebraic expressions. 3.Identify solutions of equations. 4.Translate sentences to equations. 5.Distinguish between expressions and equations. 1.2 Variables, Expressions, and Equations

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.2 – Slide Variables, Expressions, and Equations An algebraic expression is a collection of numbers, variables, operation symbols, and grouping symbols, such as parentheses, square brackets, or fraction bars. x + 5, 2m – 9, 8p 2 + 6(p – 2) Algebraic expressions A variable is a symbol, usually a letter such as x, y, or z. Different numbers can replace the variables to form specific statements. In 2m – 9, the 2m means 2 · m, the product of 2 and m; 8p 2 represents the product of 8 and p 2.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.2 – Slide 6 Evaluating Expressions 1.2 Variables, Expressions, and Equations Example 1 Find the value of each algebraic expression if m = 5 and then if m = 9. (a) 8m 8m = 8 · 5Let m = 5. Multiply.= 40 8m = 8 · 9Let m = 9. Multiply.= 72 (b) 3m 2 3m 2 = 3 · 5 2 Let m = 5. Square 5.= 3 · 25 = 75 Multiply. Let m = 9. Square 9.= 3 · 81 = 243 Multiply. 3m 2 = 3 · 9 2

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.2 – Slide Variables, Expressions, and Equations Evaluating Expressions CAUTION In example 1(b), 3m 2 means 3 · m 2 ; not 3m · 3m. Unless parentheses are used, the exponent refers only to the variable or number just before it. Use parentheses to write 3m · 3m with exponents as (3m) 2.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.2 – Slide 8 Replace x with 5 and y with 3. Example 2 Find the value of the expression if x = 5 and y = Variables, Expressions, and Equations Evaluating Expressions Multiply. Subtract. Divide.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.2 – Slide Variables, Expressions, and Equations Translating Word Phrases to Algebraic Expressions PROBLEM-SOLVING HINT Sometimes variables must be used to change word phrases into algebraic expressions. This process will be important later for solving applied problems.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.2 – Slide 10 Example 3 Write each word phrase as an algebraic expression, using x as the variable. 1.2 Variables, Expressions, and Equations Translating Word Phrases to Algebraic Expressions (a) The sum of a number and 9 “Sum” is the answer to an addition problem. This phrase translates as x + 9, or 9 + x. (b) 7 minus a number “Minus” indicates subtraction, so the translation is 7 – x. Note that x – 7 would not be correct.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.2 – Slide 11 Example 3 (continued) Write each word phrase as an algebraic expression, using x as the variable. 1.2 Variables, Expressions, and Equations Translating Word Phrases to Algebraic Expressions (c) A number subtracted from 12 Since a number is subtracted from 12, write this as 12 – x. (d) The product of 11 and a number 11 · x, or 11x (e) 5 divided by a number

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.2 – Slide 12 Example 3 (concluded) Write each word phrase as an algebraic expression, using x as the variable. 1.2 Variables, Expressions, and Equations Translating Word Phrases to Algebraic Expressions (f) The product of 2 and the difference between a number and 8. 2 (x – 8) We are multiplying 2 times another number. This number is the difference between some number and 8, written x – 8. Using parentheses around this difference, the final expression is 2(x – 8).

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.2 – Slide Variables, Expressions, and Equations CAUTION Notice that in translating the words “the difference between a number and 8” in Example 3(f), the order is kept the same: x – 8. “The difference between 8 and a number” would be written 8 – x. Translating Word Phrases to Algebraic Expressions

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.2 – Slide Variables, Expressions, and Equations Identifying Solutions of Equations To solve an equation, we must find all values of the variable that make the equation true. x +4 = 11, 2y = 16, 4p + 1 = 25 – p Equations An equation is a statement that two expressions are equal. An equation always includes the equality symbol, =. Such values of the variable are called the solutions of the equation.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.2 – Slide 15 Example 4 Decide whether the given number is a solution of the equation. 1.2 Variables, Expressions, and Equations (a) Is 7 a solution of 5p + 1 = 36? Identifying Solutions of Equations Replace p with 7. Multiply. True The number 7 is a solution of the equation.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.2 – Slide 16 Example 4 (concluded) Decide whether the given number is a solution of the equation. 1.2 Variables, Expressions, and Equations (a) Is Replace m with Multiply. False. The number Identifying Solutions of Equations a solution of 9m – 6 = 32? is not a solution of the equation.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.2 – Slide 17 Example 5 Write each word sentence as an equation. Use x as the variable. 1.2 Variables, Expressions, and Equations Translating Sentences To Equations (a) Twice the sum of a number and four is six. “Twice” means two times. The word is suggests equals. With x representing the number, translate as follows. Twice the sum of a number and four issix. 2 ·(x + 4)=6 2(x + 4) = 6

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.2 – Slide 18 Example 5 (continued) Write each word sentence as an equation. Use x as the variable. 1.2 Variables, Expressions, and Equations Translating Sentences To Equations (b) Nine more than five times a number is 49. Use x to represent the unknown number. Start with 5x and then add 9 to it. The word is translates as =. 5x + 9 = 49

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.2 – Slide 19 Example 5 (concluded) Write each word sentence as an equation. Use x as the variable. 1.2 Variables, Expressions, and Equations Translating Sentences To Equations (c) Seven less than three times a number is eleven. Here, 7 is subtracted from three times a number to get 11. Three times a number lessiseleven. 3x3x7=11 3x – 7 = 11 seven –

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.2 – Slide 20 Example 6 Decide whether each is an equation or an expression. 1.2 Variables, Expressions, and Equations Distinguishing Between Expressions and Equations (a) 2x – 5y There is no equals symbol, so this is an expression. (b) 2x = 5y Because there is an equals symbol with something on either side of it, this is an equation.