Modeling 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 the figure below.

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 \(6÷2=3\right).\)

What equation models the situation shown in the figure below? 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 \(2x=6.\)

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 \(\text{3 counters.}\) Does this check? We know \(2·3=6,\) so it works. Three counters in each of two envelopes does equal six.

The figure below 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 \(\text{12 counters.}\) How many counters are in each envelope? We have to separate the \(\text{12 counters}\) into \(\text{3 groups.}\) Since \(12÷3=4,\) there must be \(\text{4 counters}\) in each envelope. See the figure below.

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 \(3x=12.\) 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 \(3·4=12.\)

Example

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 \(\text{4 envelopes,}\) or \(4\) unknown values, on the left that match the \(\text{8 counters}\) on the right. Let’s call the unknown quantity in the envelopes \(x.\)

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

There are \(\text{2 counters}\) in each envelope.

Modeling the Division Property of Equality