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Chapter 1
Section 3
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
1.3
1
2
3
4
5
Variables, Expressions, and
Equations
Evaluate algebraic expressions, given values
for the variables.
Translate word phrases to algebraic
expressions.
Identify solutions of equations.
Identify solutions of equations from a set of
numbers.
Distinguish between expressions and
equations.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Variables, Expressions, and Equations
A variable is a symbol, usually a letter such as x, y, or z, used
to represent any unknown number.
An algebraic expression is a sequence of numbers, variables
operation symbols and/or grouping symbols (such as parentheses)
formed according to the rules of algebra.
x  5, 2m  9, 8 p2  6  p  2 Algebraic expressions
In 2m  9, the 2m means 2  m , the product of 2 and m; 8p2
represents the product of 8 and p2. Also, 6( p  2) means the
product of 6 and p  2 .
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1.3- 3
Objective 1
Evaluate algebraic expressions,
given values for the variables.
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Slide 1.3- 4
EXAMPLE 1
Evaluating Expressions
Find the value of each algebraic expression when p  3.
Solution:
16 p  16  3  48
2 p 3  2  33  2  27  54
Remember, 2p3 means 2 · p3, not 2p· 2p · 2p. Unless
parentheses are used, the exponent refers only to the variable
or number just before it. To write 2p· 2p · 2p with exponents, use
(2p)3.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1.3- 5
EXAMPLE 2
Evaluating Expressions
Find the value of each expression when x  6 and
y  9.
Solution:
4 x  5 y  4  6  5  9  24  45  69
4x  2 y
46  29
24  18 6



x 1
6 1
7
7
2x  y  2  6  9  2  36  81  72  81  153
2
2
2
2
A sequence such as 3) · x ( + y is not an algebraic expression
because the rules of algebra require a closing parentheses or
bracket for every opening parentheses or bracket
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Slide 1.3- 6
Objective 2
Translate word phrases to
algebraic expressions.
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Slide 1.3- 7
EXAMPLE 3
Using Variables to Write Word
Phrases as Algebraic Expressions
Write each word phrase as an algebraic expression using x
as the variable.
A number subtracted from 48
Solution: 48  x
The product of 6 and a number
6x
9 multiplied by the sum of a number and 5
9  x  5
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1.3- 8
Objective 3
Identify solutions of equations.
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Slide 1.3- 9
Identify solutions of equations.
An equation is a statement that two algebraic
expressions are equal. Therefore, an equation always
includes the equality symbol, = .
x  4  11 , 2 y  16 , 4 p  1  25  p
3
1
z  4 , x   0 , 4  m  0.5  2m
4
2
2
}
Equations
To solve an equation means to find the values of the
variable that make the equation true. Such values of the
variable are called the solutions of the equation.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1.3- 10
EXAMPLE 4
Deciding whether a Number Is a
Solution of an Equation
Decide whether the given number is a solution of the
equation.
Solution:
8 p  11  5;2
8  2 11  5
16 11  5
55
Yes
Remember that the rules of operations still apply to equations.
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Slide 1.3- 11
Objective 4
Identify solutions of equations
from a set of numbers.
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Slide 1.3- 12
Identify solutions of equations from a
set of numbers.
A set is a collection of objects. In mathematics these
objects are most often numbers. The objects that belong
to the set, called elements of the set, are written
between braces. For example, the set containing the
numbers (or elements) 1, 2, 3, 4, and 5 is written as
{1, 2, 3, 4, 5}.
One way of determining solutions is the direct
substitution of all possible replacements. The ones that
lead to true statements are solutions.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1.3- 13
EXAMPLE 5
Finding a Solution from a Given
Set
Write the statement as an equation. Find all solutions from the
set {0, 2, 4, 6, 8, 10}.
Three times a number is subtracted from 21, giving 15.
Solution:
21  3x  15
2 is the solution from this set of elements.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1.3- 14
Objective 5
Distinguish between expressions
and equations.
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Slide 1.3- 15
Distinguish between equations and
expressions.
An equation is a sentence—it has something on the
left side, an = sign, and something on the right side.
An expression is a phrase that represents a number.
4x  5  9
4x  5
Equation
Expression
(to solve)
(to simplify or evaluate)
One way to help figure this out is, equation and equal are similar.
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Slide 1.3- 16
EXAMPLE 6
Distinguishing between
Equations and Expressions
Decide whether the following is an equation or an
expression.
3x  1
5
Solution:
There is no equals sign, so this
is an expression.
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Slide 1.3- 17