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Evaluating Variable Expressions

2.1.1 Evaluating Variable Expressions

Variable, Term and Coefficient

A variable is a letter used to represent a quantity that is unknown,
or a quantity that can change or vary.


x = the price of one share of Microsoft stock
y = the cost of a new car
h = a student’s height
m = miles per gallon of an automobile

The terms of a variable expression are the addends of the expression. Consider

a. Rewrite the expression using only addends.

b. List the variable terms

c. List the constant term.

d. What are factors?

e. How many factors are in the first term? List them.

f. How many factors are in the second term? List them

The numerical coefficients of an expression, or simply the
coefficients, are the numerical factors of the variable terms.

g. List the coefficients in

Evaluate an Expression

To evaluate a variable expression means to replace the variables
with numbers and perform all indicated operations (addition,
subtraction, multiplication, division, etc); the result will be a single

Rule on Substituting Negative Numbers into Variables

Whenever a negative number is substituted into a variable, the
negative number must be placed inside a set of parentheses.
Furthermore, if a negative number is raised to an exponent, the
negative number must be inside a set of parentheses and the
exponent must be outside the parentheses.

Example 1 Evaluate x^2 when


Example 2 Evaluate −b^2 when


Example 3 Evaluate −y^2 − 3xy when


Example 4 Evaluate



Example 5 Evaluate when


Example 6 Evaluate


2.2.1 Properties of Real Numbers

If a, b, and c are real numbers, then

Property Addition Multiplication

means order does
not matter

Commutative property of
Commutative property of
means grouping
does not matter
Associative property of
Associative property of
The additive inverse
is the opposite. The
inverse is the
Inverse property of
Inverse property of
Zero Addition property of
Multiplication property
of zero
One   Multiplication property
of one

The Distributive Property

For any real numbers a, b, and c
a(b + c) = ab + ac and (b + c)a = ab + ac

Example 1

Rewrite each expression using the distributive
property. Then evaluate the expression.


Example 2 Identify the property that justifies each statement

a. 12 + (−12) = 0

b. (3+11)+15 = 3+ (11+15)

c. (4 * 7) *5 = (7 * 4) *5

d. Any number times its reciprocal equals one.

e. Any number times zero is zero.

f. The sum of any number and its additive inverse is zero.

g. The order in which two numbers are multiplied does not matter.

Example 3 Create an example that shows division does not
have the commutative property.

Example 4 Create an example that shows subtraction does not
have the associative property.

2.2.2 Combining Like Terms

Terms are addends. Like Terms of an expression are terms that
have identical variable parts. Constant terms are always like terms.
To combine like terms means to add the coefficients of like terms,
keeping the variable part the same. This is possible because of the
distributive property. For example,

Example 5 Use the distributive property to simplify each expression.

Example 6 Simplify each expression

2.2.3 Use the Properties of Multiplication to Simplify
Variable Expressions

Example 7 Use the Commutative and Associative Properties of
Multiplication to simplify each expression.

a. −4(c * 7)

−4(c * 7) = −4(7c) Commutative property of multiplication
= (−4 * 7)c Associative property of multiplication
= −28c  

Example 8 Simplify each expression

2.2.4 Simplify Variable Expressions using the
Distributive Property

Example 9 Simplify each expression.

2.2.5 Simplify General Expressions

Steps to Simplify General Expressions

1. Use the Distributive Property to remove any grouping symbols starting from the innermost set and working outward.
2. Combine like terms.
3. Write the terms in alphabetical order with the constant term last.

Example 10 Simplify each expression

2.3 Translating Verbal Expressions into Variable

In the expressions below marked with an asterisk (*), the actual
operation (+, −, ×, ÷ ) occurs at the word “and” in the sentence.

  Verbal Expression Variable Expression
6 added to y y + 6
8 more than x x + 8
*the sum of x and z x + z
t increased by 9 t + 9
*the total of 5 and y 5 + y
x minus two
seven less than t
x − 2
t − 7
seven less t
5 subtracted from d
*the difference between y and 4
m decreased by 3
7 − t
d − 5
y − 4
m − 3
10 times t 10t
one-half of x 1/2 x
*the product of y and z yz
11 multiplied by y 11y
twice n 2n
x divided by 12
the ratio of t to nine
*the quotient of y and z
Powers the square of x
the cube of z

Example 1 Translate each into a variable expression.

a. The total of five times b and c.


b. Five times the total of b and c.


c. The quotient of eight less than n and fourteen.


d. Thirteen more than the sum of seven and the square of x.


e. Eighteen less than the cube of x.


f. Eighteen less the square of x


g. y decreased by the sum of z and nine.


h. The difference between the square of q and the sum of r and t.


2.3.2 Translate into a variable expression

Example 2 Translate each of the following into a variable

a. Translate “a number multiplied by the total of six and the cube
of the number.”


b. Translate “a number added to the product of five and the
square of the number.”


c. Translate “the quotient of twice a number and the difference
between the number and twenty.”


d. Translate “the product of three and the sum of seven and twice
a number.”


2.3.3 Translate into a variable expression, and then
simplify the variable expression.

Example 3 Translate and simplify each expression.

a. Translate and simplify “the total of four times a number and
twice the difference between the number and eight.”


b. Translate and simplify “a number minus the difference
between twice the number and seventeen.”


c. Translate and simplify “the difference between five eighths of
a number and two thirds of the number.”


d. Translate and simplify “the sum of three fourths of a number
and one fifth of the number.”


2.3.4 Defining Variables

Example 4
For each of the following define a variable and
express all unknown quantities in terms of that

a. The length of a swimming pool is 20 feet longer than the
width. Express the length of the pool in terms of the width.


b. An older computer takes twice the amount of time to process
data as a new computer. Express the amount of time it takes
the older computer to process the data in terms of the amount
of time it would take the new computer.


c. A guitar string 6 ft long was cut into two pieces. Use one
variable to express the lengths of the two pieces.

d. If the sum of two numbers is 10. Use one variable to express
each number.


e. An investor divided $5000 into two accounts, one a mutual
fund and the other a money market fund. Use one variable to
express the amounts invested in each account.