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Solve:

y + 15 = −4

−19

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Solve: a 6 = −8

Solution

.
Add 6 to each side to undo the subtraction. .
Simplify. .

Check the result by substituting −2 into the original equation: a 6 = −8 .

Substitute −2 for a −2 6 = ? −8
−8 = −8

The solution to a 6 = −8 is −2 .

Since a = −2 makes a 6 = −8 a true statement, we found the solution to this equation.

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Solve:

a 2 = −8

−6

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Solve:

n 4 = −8

−4

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Model the division property of equality

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. Now we’ll see how to solve equations that involve division.

We will model an equation    with envelopes and counters in [link] .

This image has two columns. In the first column are two identical envelopes. In the second column there are six blue circles, randomly placed.

Here, 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?

To determine the number, separate the counters on the right side into 2 groups of the same size. So 6 counters divided into 2 groups means there must be 3 counters in each group (since 6 ÷ 2 = 3 ) .

What equation models the situation shown in [link] ? There are two envelopes, and each contains x counters. Together, the two envelopes must contain a total of 6 counters. So the equation that models the situation is 2 x = 6 .

This image has two columns. In the first column are two identical envelopes. In the second column there are six blue circles, randomly placed. Under the figure is two times x equals 6.

We can divide both sides of the equation by 2 as we did with the envelopes and counters.

This figure has two rows. The first row has the equation 2x divided by 2 equals 6 divided by 2. The second row has the equation x equals 3.

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.

[link] shows another example.

This image has two columns. In the first column are three envelopes. In the second column there are four rows of  three blue circles. Underneath the image is the equation 3x equals 12.

Now we have 3 identical envelopes and 12 counters. How many counters are in each envelope? We have to separate the 12 counters into 3 groups. Since 12 ÷ 3 = 4 , there must be 4 counters in each envelope. See [link] .

This image has two columns. In the first column are four envelopes. In the second column there are twelve blue circles.

The equation that models the situation is 3 x = 12 . We can divide both sides of the equation by 3 .

This image shows the equation 3x divided by 3 equals 12 divided by 3. Below this equation is the equation x equals 4.

Does this check? It does because 3 · 4 = 12 .

Doing the Manipulative Mathematics activity “Division Property of Equality” will help you develop a better understanding of how to solve equations using the Division Property of Equality.

Write an equation modeled by the envelopes and counters, and then solve it.

This image has two columns. In the first column are four envelopes. In the second column there are 8 blue circles.

Solution

There are 4 envelopes, or 4 unknown values, on the left that match the 8 counters on the right. Let’s call the unknown quantity in the envelopes x .

Write the equation. .
Divide both sides by 4. .
Simplify. .

There are 2 counters in each envelope.

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Write the equation modeled by the envelopes and counters. Then solve it.
This image has two columns. In the first column are four envelopes. In the second column there are 12 blue circles.

4 x = 12; x = 3

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Write the equation modeled by the envelopes and counters. Then solve it.
This image has two columns. In the first column are three envelopes. In the second column there are six blue circles.

3 x = 6; x = 2

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Solve equations using the division property of equality

The previous examples lead to the Division Property of Equality . When you divide both sides of an equation    by any nonzero number, you still have equality.

Division property of equality

For any numbers a , b , c , and c 0

If a = b then a c = b c .

Solve: 7 x = −49 .

Solution

To isolate x , we need to undo multiplication.

.
Divide each side by 7. .
Simplify. .

Check the solution.

7 x = −49
Substitute −7 for x. 7 ( −7 ) = ? −49
−49 = −49

Therefore, −7 is the solution to the equation.

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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