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Modeling the Subtraction Property of Equality

Modeling the Subtraction Property of Equality

We will use a model to help you understand how the process of solving an equation is like solving a puzzle. An envelope represents the variable – since its contents are unknown – and each counter represents one.

Suppose a desk has an imaginary line dividing it in half. We place three counters and an envelope on the left side of desk, and eight counters on the right side of the desk as in the figure below. Both sides of the desk have the same number of counters, but some counters are hidden in the envelope. Can you tell how many counters are in the envelope?

The image is divided in half vertically. On the left side is an envelope with three counters below it. On the right side is 8 counters.

What steps are you taking in your mind to figure out how many counters are in the envelope? Perhaps you are thinking “I need to remove the \(3\) counters from the left side to get the envelope by itself. Those \(3\) counters on the left match with \(3\) on the right, so I can take them away from both sides. That leaves five counters on the right, so there must be \(5\) counters in the envelope.” The figure below shows this process.

The image is in two parts. On the left is a rectangle divided in half vertically. On the left side of the rectangle is an envelope with three counters below it. The 3 counters are circled in red with an arrow pointing out of the rectangle. On the right side is 8 counters. The bottom 3 counters are circled in red with an arrow pointing out of the rectangle. The 3 circled counters are removed from both sides of the rectangle, creating the new rectangle on the right of the image which is also divided in half vertically. On the left side of the rectangle is just an envelope. On the right side is 5 counters.

What algebraic equation is modeled by this situation? Each side of the desk represents an and the center line takes the place of the equal sign. We will call the contents of the envelope \(x,\) so the number of counters on the left side of the desk is \(x+3.\) On the right side of the desk are \(8\) counters. We are told that \(x+3\) is equal to \(8\) so our equation is\(x+3=8.\)

The image is divided in half vertically. On the left side is an envelope with three counters below it. On the right side is 8 counters.

\(x+3=8\)

Let’s write algebraically the steps we took to discover how many counters were in the envelope.

.
First, we took away three from each side. .
Then we were left with five. .

Now let’s check our solution. We substitute \(5\) for \(x\) in the original equation and see if we get a true statement.

The image shows the original equation, x plus 3 equal to 8. Substitute 5 in for x to check. The equation becomes 5 plus 3 equal to 8. Is this true? The left side simplifies by adding 5 and 3 to get 8. Both sides of the equal symbol are 8.

Our solution is correct. Five counters in the envelope plus three more equals eight.

Example

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

The image is divided in half vertically. On the left side is an envelope with 4 counters below it. On the right side is 5 counters.

Solution

On the left, write \(x\) for the contents of the envelope, add the \(4\) counters, so we have \(x+4\). \(x+4\)
On the right, there are \(5\) counters. \(5\)
The two sides are equal. \(x+4=5\)
Solve the equation by subtracting \(4\) counters from each side.
The image is in two parts. On the left is a rectangle divided in half vertically. On the left side of the rectangle is an envelope with 4 counters below it. The 4 counters are circled in red with an arrow pointing out of the rectangle. On the right side is 5 counters. The bottom 4 counters are circled in red with an arrow pointing out of the rectangle. The 4 circled counters are removed from both sides of the rectangle, creating the new rectangle on the right of the image which is also divided in half vertically. On the left side of the rectangle is just an envelope. On the right side is 1 counter.

We can see that there is one counter in the envelope. This can be shown algebraically as:

The image shows the given equation, x plus 4 equal to 5. Take 4 away from both sides of the equation to get x plus 4 minus 4 equal to 5 minus 4. On the left, plus 4 and minus 4 cancel out to leave just x. On the right 5 minus 4 is 1. The equation becomes x equal to 1.

Substitute \(1\) for \(x\) in the equation to check.

The image shows the original equation, x plus 4 equal to 5. Substitute 1 in for x to check. The equation becomes 1 plus 4 equal to 5. Is this true? The left side simplifies by adding 1 and 4 to get 5. Both sides of the equal symbol are 5.

Since \(x=1\) makes the statement true, we know that \(1\) is indeed a solution.

This lesson is part of:

The Language of Algebra

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