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2.2.4 The Addition Property of Equality
One of the most important laws in algebra is also one of the most simple. The Addition Property of Equality tells us how numbers can be added to and subtracted from equations. The example below shows how it's used.
 Example 2.6: Mike and Jane are on opposite sides in a pickup volleyball game. Naturally both teams have the same number of players. All of the players have about the same ability and the game is fair and competitive. A half hour into the game, four new players show up and ask to get in the game. Naturally, the new group of players is split and the same number of players (2) is added to each side. Since the same number of players is added to each team, the game stays fair and competitive.
In the pickup game,
m= the number of players on Mike's team
j= the number of players on Jane's team
n = the number of new players added to each team
Since Mike and Jane have the same number of players on their teams, we can write the equation
m = j
Since they add the same number of new players, n, to their teams, we can write
m + n = j + n
This is an example of the Addition Property of Equality. It basically says that if you have an equation, it will stay equal if you add the same amount to both sides. In math, it’s stated like this:
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The Addition Property of Equality
For any real numbers a,b, and c
if a = b, then
a + c = b + c
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The property also holds true for subtraction; the equation stays equal if you subtract the same amount from both sides. In the volleyball game, the teams would have stayed equal if two players on each team had left the game. While this concept may seem too simple to have its own property, it is one of algebra's most important tools. The property is used over and over to solve algebra problems.
The following example shows how the Addition Property of Equality is used.
 Example 2.7: On the second down, the Wildcats are on their own 21 yard line. On the third down they're on their own 38 yard line. How far did they move the ball on the second down?
Solution: For this problem, we can define the terms as follows:
s = the field position on the second down = 21
t = the field position on the third down = 38
m = the number yards moved on the second down
The terms are related by following equation
t = s + m
or 38 = 21 + m
m + 21 = 38
If we want to find the value of m, we must get the equation to look like:
m = ____
The key to getting the equation into this form is to get rid of the 21 on the left side of the last equation. We can do this by subtracting 21 from (or adding 21 to) each side.
The Wildcats moved the ball 17 yards on the second down.
Note - In this example we used another very simple but important law of mathematics, the Additive Identity Property, when we said m + 0 = m.
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The Additive Identity Property
For any real number a,
a + 0 =a
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