CHAPTER 5 Solving First Degree Equations in One Variable
CHAPTER 5 Solving First Degree Equations in One Variable
5.2 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 need to be simplified
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 or . 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 1).
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 ).
What equation models the situation shown in (Figure 2)? There are two envelopes, and each contains counters. Together, the two envelopes must contain a total of 6 counters.
| 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 , so it works! Three counters in each of two envelopes does equal six!
This example leads to the Division Property of Equality.
Division and Multiplication Properties of Equality
Division Property of Equality: For all real numbers , and , if , then .
Multiplication Property of Equality: For all real numbers , if , then .
When you divide or multiply both sides of an equation by the same quantity, you still have equality.
Let’s review how these properties of equality can be applied in order to solve equations. Remember, the goal is to ‘undo’ the operation on the variable. In the example below the variable is multiplied by , so we will divide both sides by to ‘undo’ the multiplication.
EXAMPLE 1
Solve: .
Solution
We use the Division Property of Equality to divide both sides by .
| Divide both sides by 4 to undo the multiplication. | |
| Simplify. | |
| Check your answer. Let . |
|
Since this is a true statement, is a solution to .
TRY IT 1.1
Solve: .
Show answer
y = −16
TRY IT 1.2
Solve: .
Show answer
z = −13
In the previous example, to ‘undo’ multiplication, we divided. How do you think we ‘undo’ division?
EXAMPLE 2
Solve: .
Solution
Here is divided by . We can multiply both sides by to isolate .
| Multiply both sides by . | |
| Simplify. | |
| Check yo