19

2.2: Solving Equations by Multiplying and Dividing
Objectives:
2.2.1:
Use the multiplication property to solve
equations
2.2.2:
Use the multiplication property to solve an
application
20

The Multiplication Property of Equality
Suppose we want to solve this equation: 6x = 18.
The addition property doesn’t help to solve the equation.
21

2.2.1: Solving Equations by Using the Multiplication Property
Examples: Solve the following equations
1.
6x = 18
2.
5x =
–
35
3.
–
9x = 54
4.
= 6
5.
=
–
9
3
x
5
x
22

2.2.1: Solving Equations by Using the Multiplication Property
Examples: Solve the following equations
Solving Equations by
Using Reciprocals
1.
x = 9
2.
x
= 18
Solving Equations by
Combining Like Terms
1.
3x + 5x = 40
2.
7x + 4x =
–
66
5
3
3
2
23

2.2.2: Use the Multiplication Property to Solve an Application
Example:
On her first day on the job in a photography lab, Nancy
processed all of the film given to her. The following day, her
boss gave her four times as much film to process. Over the
two days, she processed 60 rolls of film. How many rolls did
she process on the first day?
24

2.3: Combining the Rules to Solve Equations
Objectives:
2.3.1: Use both addition and multiplication to solve
equations
2.3.2: Solve equations involving fractions
2.3.3: Solve applications
25

2.3.1: Use Both Addition and Multiplication to Solve Equations
Examples: Solve the following equations
1.
3x
–
5 = 4
2.
4x
–
7 = 17
3.
5x
–
11 = 2x
–
7
4.
7x
–
12 = 2x
–
9
26

2.3.1: Use Both Addition and Multiplication to Solve Equations
Examples: Solve the following equations
Applying the Properties of Equality with
Like Terms
1.
8x + 2
–
3x = 8 + 3x + 2
2.
7x
–
3
–
5x = 10 + 4x + 3
Applying the Properties of Equality with
Parentheses
1.
x + 3(3x
–
1) = 4(x + 2) + 4
2.
x + 5(x + 2) = 3(3x
–
2) + 18
27

2.3.2: Solve Equations Involving Fractions
To solve an equation involving fractions, the first step is to
multiply both sides of the equation by the
least common
multiple (LCM)
of all denominators in the equation.
This clears the equation of fractions, and we can proceed as
before.
The LCM of a set of denominators is also called the
least
common denominator (LCD).
Examples: Solve the following equations
1.
–
=
2.
+ 1 =
2
x
3
2
6
5
5
1
2
x
2
x
28

Conditional Equations, Identities, and Contradictions
An equation that is
true for only particular
values of the
variable is called a
conditional equation
.
Here the
equation can be written in the form
ax + b = 0
in which a ≠ 0
An equation that
is true
for
all
possible values of the
variable is called an
identity
.
In this case,
both a
and
b
are
0, so we get the equation 0 = 0. This is the case, if both
sides of the equation reduce to the same expression (a true
statement).
An equation that is
never true
, no matter what the value of
the variable, is called a
contradiction
.
For example, if
a
is
0 but
b
is 4, a contradiction results. This is the case if the
equation simplifies as a false statement
29

Identities and Contradictions
Examples: Solve the following equations
Solving
Identity Equation
1.

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- Fall '18
- jane
- Accounting, Linear Equations, Elementary algebra