copyright © 2010 pearson education, inc. all rights reserved. 1.2 – slide 1


The Real Number System

Chapter 1


1.2

Variables, Expressions, and

Equations


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



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, 8p2 + 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;8p2 represents the product of 8 and p2.


Evaluating Expressions


Example 1

Find the value of each algebraic expression if m = 5 and then if m = 9.

(a) 8m8m = 8 · 5 Let m = 5.

Multiply.= 40

8m = 8 · 9 Let m = 9.

Multiply.= 72

(b) 3m2

3m2 = 3 · 52 Let m = 5.

Square 5.= 3 · 25

= 75 Multiply.

Let m = 9.

Square 9.= 3 · 81

= 243 Multiply.

3m2 = 3 · 92


1.2 Variables, Expressions, and EquationsEvaluating Expressions

CAUTION

In example 1(b), 3m2 means 3 · m2; 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.


9 8(b)

2

y

y

x

x

Replace x with 5 and y with 3.

45 24

10 3

Example 2

Find the value of the expression if x = 5 and y = 3.

1.2 Variables, Expressions, and EquationsEvaluating Expressions

Multiply.

21

7 Subtract.

3 Divide.

95 8325 3


1.2 Variables, Expressions, and EquationsTranslating 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.


Example 3

Write each word phrase as an algebraic expression, using x as the variable.


(a) The sum of a number and 9“Sum” is the answer to an addition problem. This phrasetranslates 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.


Example 3 (continued)



(c) A number subtracted from 12Since 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 number5

5 , or xx


Example 3 (concluded)



(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).



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


1.2 Variables, Expressions, and EquationsIdentifying 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.


Example 4

Decide whether the given number is a solution of the equation.


(a) Is 7 a solution of 5p + 1 = 36?

5 1 36p

5 67 1 3

Identifying Solutions of Equations

Replace p with 7.

35 1 36 Multiply.36 36 True

The number 7 is a solution of the equation.



Decide whether the given number is a solution of the equation.


(a) Is

Replace m with

Multiply.False.

The number

Identifying Solutions of Equations

14

3

9 6 32m 14

9 6 323

42 6 32 36 32

14

3

a solution of 9m – 6 = 32?

is not a solution of the equation.

14.

3


Example 5

Write each word sentence as an equation. Use x as the variable.

1.2 Variables, Expressions, and EquationsTranslating 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.

Twicethe sum of a

number and fouris six.

2 · (x + 4) = 6

2(x + 4) = 6


Example 5 (continued)



(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





(c) Seven less than three times a number is eleven.Here, 7 is subtracted from three times a number to get 11.

Three timesa number less is eleven.

3x 7 = 11

3x – 7 = 11

seven

–


Example 6

Decide whether each is an equation or an expression.

1.2 Variables, Expressions, and EquationsDistinguishing 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.

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