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2.1 Evaluating Variable Expressions (Page 1 of 14)
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.
e.g.
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 2x 3  4xy  2y2  7
a. Rewrite the expression using
only addends (terms).
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
2x 3  4xy  2y2  7
2.1 Evaluating Variable Expressions (Page 2 of 14)
Evaluating a Variable 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
number.
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
a. x  7
Evaluate x 2 when
b. x  4
c. x  5
Example 2
a. b  4
Evaluate b 2 when
b. b  7
c. b  5
Example 3
Evaluate y2  3xy when
a. x  4 and y  2
b. x  5 and y  3
2.1 Evaluating Variable Expressions (Page 3 of 14)
a2  b2
Example 4
Evaluate
when
ab
a. a  3 and b  4
b. a  3 and b  5 .
Example 5
Evaluate x 2  2(x  y)  z 3 when
a. x  2 , y  2 , z  3
b. x  2 , y  2 , z  3
Example 6
Find the volume of a right
circular cylinder that has radius
1.25 in. and a height of 5.25 in.
Round to the nearest tenth of a
1
cubic inch. Use V   r 2 h and
3
  3.14 .
2.2 Properties of Real Numbers (Page 4 of 14)
2.2.1 Properties of Real Numbers
If a, b, and c are real numbers, then
Property
Operation
Addition
Multiplication
Commutative
means order does
not matter
Commutative
property of
addition
Commutative
property of
multiplication
Associative
means grouping
does not matter
Associative
property of
addition
Associative
property of
multiplication
Inverse property
of addition
Inverse property
of multiplication
Addition property
of zero
Multiplication
property of zero
Inverse
The additive inverse
is the opposite. The
multiplicative
inverse is the
reciprocal.
Zero
One
Multiplication
property of one
2.2 Properties of Real Numbers (Page 5 of 14)
The Distributive Property
For any real numbers a, b, and c
a(b  c)  ab  ac and (b  c)a  ab  ac
Example 1
a. 3(4  5)
Rewrite each expression using the distributive
property. Then evaluate the expression.
b. (3 5) 4
solution
3(4  5)  3 4  (3) 5
 12  (15)
 27
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.
2.2 Properties of Real Numbers (Page 6 of 14)
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 (see example).
3y  4 y  (3  4)y
a. 4x  7x
Use the distributive property to simplify each
expression.
b. 22a  31a
Example 6
a. 18b  4b
Simplify each expression.
b. 6t  5t
Example 5
c. 4x  13  5y  3x  12  2y
 (7)y
 7y
d. 3a 2  2a  5  a 2  16
2.2 Properties of Real Numbers (Page 7 of 14)
2.2.3 Use the Properties of Multiplication to Simplify
Variable Expressions
Example 7
a. 4(c  7)
Use the Commutative and Associative Properties of
Multiplication to simplify each expression.
4(c  7)  4(7c)
 (4  7)c
 28c
b. 2(5x)
Example 8
a.
4(x)
Commutative property of
multiplication
Associative property of
multiplication
c. 2[a(15)]
Simplify each expression
b.
14(5c)
c.
3 2 
  y
2 3 
d.
5
 (36a 2 )
6
e.
16(3x)
f.
5a(7)
2.2 Properties of Real Numbers (Page 8 of 14)
2.2.4
Simplify Variable Expressions using the
Distributive Property
Example 9
Simplify each expression.
a.
b.
3(4x  5)
2(5x  9)
c.
2(3x  7)
d.
7(2x  5)
e.
3(4  6x)
f.
(3x 10)
g.
(3y  4)7
h.
(2x  8) 6
i.
4(6b 2  4b  7)
j.
5(3x 3  7x  9)
2.2 Properties of Real Numbers (Page 9 of 14)
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.
a. 4(x  y)  2(2x  5y)
b. 7(a  4b)  2(2a  3b)
c. 3(2x  y)  (3x  4y)
d. 3(2a  9b)  (3a  4b)
e. 5x  3[2x  6(x  7)]
f. 3a  6[a  5(2  4a)]
2.3 Translating Verbal Expressions (Page 10 of 14)
2.3
Translating Verbal Expressions into Variable
Expressions
In the expressions below marked with an asterisk (*), the actual
operation (+, , ,  ) occurs at the word “and” in the sentence.
Verbal Expression
Addition
Phrases
Subtraction
Phrases
Multiplication
Phrases
Division
Phrases
Powers
6 added to y
8 more than x
*the sum of x and z
t increased by 9
*the total of 5 and y
x minus two
seven less than t
5 subtracted from d
*the difference between
y and 4
m decreased by 3
10 times t
one-half of x
*the product of y and z
11 multiplied by y
twice n
x divided by 12
the ratio of t to nine
*the quotient of y and z
the square of x
the cube of z
Variable Expression
y6
x8
xz
t9
5 y
x2
t7
d5
y4
m3
10t
1
2 x
yz
11y
2n
x
12
t
9
y
z
2
x
z3
2.3 Translating Verbal Expressions (Page 11 of 14)
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 Translating Verbal Expressions (Page 12 of 14)
2.3.2
Translate into a variable expression
Example 2
a. Translate “a number
multiplied by the total of six
and the cube of the number”
into a variable expression.
b. Translate “a number added
to the product of five and the
square of the number” into a
variable expression.
c. Translate “the quotient of
twice a number and the
difference between the
number and twenty” into a
variable expression.
d. Translate “the product of
three and the sum of seven
and twice a number” into a
variable expression.
2.3 Translating Verbal Expressions (Page 13 of 14)
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 Translating Verbal Expressions (Page 14 of 14)
2.3.4
Defining Variables
Example 4 For each of the following (1) define a variable, and
(2) express all unknown quantities in terms of that
variable.
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.