Solve Equations using the Division and Multiplication Properties of
Equality
Learning Objectives
By the end of this section, you will be able to:
Solve equations using the Division and Multiplication Properties of Eq
...
Solve Equations using the Division and Multiplication Properties of
Equality
Learning Objectives
By the end of this section, you will be able to:
Solve equations using the Division and Multiplication Properties of Equality
Solve equations that require simplification
Translate to an equation and solve
Translate and solve applications
Be Prepared!
Before you get started, take this readiness quiz.
1. Simplify: −7⎛ ⎝−7 1 ⎞ ⎠.
If you missed this problem, review Example 1.68.
2. Evaluate 9x + 2 when x = −3 .
If you missed this problem, review Example 1.57.
Solve Equations Using the Division and Multiplication Properties of Equality
You may have noticed that all of the equations we have solved so far have been of the form x + a = b or x − a = b . We
were able to isolate the variable by adding or subtracting the constant term on the side of the equation with the variable.
Now we will see how to solve equations that have a variable multiplied by a constant and so will require division to isolate
the variable.
Let’s look at our puzzle again with the envelopes and counters in Figure 2.5.
Figure 2.5 The illustration shows a model
of an equation with one variable multiplied
by a constant. On the left side of the
workspace are two instances of the
unknown (envelope), while on the right side
of the workspace are six counters.
In the illustration there are two identical envelopes that contain the same number of counters. Remember, the left side
of the workspace must equal the right side, but the counters on the left side are “hidden” in the envelopes. So how many
counters are in each envelope?
How do we determine the number? We have to separate the counters on the right side into two groups of the same size
to correspond with the two envelopes on the left side. The 6 counters divided into 2 equal groups gives 3 counters in each
group (since 6 ÷ 2 = 3 ).
What equation models the situation shown in Figure 2.6? There are two envelopes, and each contains x counters.
Together, the two envelopes must contain a total of 6 counters.
212 Chapter 2 Solving Linear Equations and Inequalities
This OpenStax book is available for free at http://cnx.org/content/col12116/1.2Figure 2.6 The illustration shows a model
of the equation 2x = 6 .
If we divide both sides of the equation by 2, as we did with the envelopes and counters,
we get:
We found that each envelope contains 3 counters. Does this check? We know 2 · 3 = 6 , so it works! Three counters in
each of two envelopes does equal six!
This example leads to the Division Property of Equality.
The Division Property of Equality
For any numbers a, b, and c, and c ≠ 0 ,
If a = b,
then a
c = bc
When you divide both sides of an equation by any non-zero number, you still have equality.
MANIPULATIVE MATHEMATICS
Doing the Manipulative Mathematics activity “ Division Property of Equality” will help you develop a better
understanding of how to solve equations by using the Division Property of Equality.
The goal in solving an equation is to ‘undo’ the operation on the variable. In the next example, the variable is multiplied
by 5, so we will divide both sides by 5 to ‘undo’ the multiplication.
EXAMPLE 2.13
Solve: 5x = −27.
Solution
To isolate x , “undo” the multiplication by 5.
Divide to ‘undo’ the multiplication.
Simplify.
Check:
Chapter 2 Solving Linear Equations and Inequalities 213Substitute −27
5 for x.
Since this is a true statement, x = − 27
5
is the solution to 5x = −27 .
TRY IT : : 2.25 Solve: 3y = −41.
TRY IT : : 2.26 Solve: 4z = −55.
Consider the equation x
4 = 3 . We want to know what number divided by 4 gives 3. So to “undo” the division, we will need
to multiply by 4. The Multiplication Property of Equality will allow us to do this. This property says that if we start with
two equal quantities and multiply both by the same number, the results are equal.
The Multiplication Property of Equality
For any numbers a, b, and c,
If a = b,
then ac = bc
If you multiply both sides of an equation by the same number, you still have equality.
EXAMPLE 2.14
Solve: y
−7 = −14.
Solution
Here y is divided by −7 . We must multiply by −7 to isolate y .
Multiply both sides by −7 .
Multiply.
Simplify.
Check: y
−7 = −14
Substitute y = 98 .
Divide.
TRY IT : : 2.27 Solve: a
−7 = −42.
TRY IT : : 2.28
Solve: b
−6 = −24.
214 Chapter 2 Solving Linear Equations and Inequalities
This OpenStax book is available for free at http://cnx.org/content/col12116/1.2EXAMPLE 2.15
Solve: −n = 9.
Solution
Remember −n is equivalent to −1n .
Divide both sides by −1 .
Divide.
Notice that there are two other ways to solve −n = 9 . We can also solve this equation by multiplying both
sides by −1 and also by taking the opposite of both sides.
Check:
Substitute n = −9 .
Simplify.
TRY IT : : 2.29 Solve: −k = 8.
TRY IT : : 2.30 Solve: −g = 3.
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