Introductory Algebra

1.2 – The Addition Principle of Equations

The Addition Principle of Equalities

Imagine we have a fulcrum and a lever with two objects, one on each end of the lever. The objects are shaped differently, but have the same weight. The lever is in balance.

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Now imagine a third object is placed at one side of the lever, but not on the other. What happens?

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Our lever falls out of balance. The weights are no longer equal. But if we were to add the same weight to both sides …

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We see that we have maintained a balance again, often called an equilibrium.

The same concept can be applied to mathematical equations. In fact, the fulcrum and lever balancing act above is a very important concept in physics, and is modeled by mathematical equations. We will, of course, focus on the mathematical aspects.

 

Addition Principle of Equalities
The Addition Principle of Equalities states that if you add the same value to each side of an equation, you maintain the equality. Mathematically, this is:
If a=b

Then a+c=b+c

Or in other words, if we add the same thing to each side of an equation, we maintain the equation. This can be particularly useful in equations such as:

 

x-7=8
We can add a number to both sides of this equation. If we choose wisely, we might eliminate the constant and isolate the variable. Let’s add the opposite of (-7) to both sides:
x-7=8
x-7{\color{red}+7}=8{\color{red}+7}

Simplifying, this becomes
x+0=15

And adding by zero leaves x unchanged, so we write:
x=15

And we have solved this equation! We may wish to check our solution by replacing x with 15 in our original problem.
x-7=8

15-7=8

8=8

Which is true

 

Solve the equation: x-4=8
Again, we begin by adding the opposite of the constant to each side:
x-4=8

x-4{\color{red}+4}=8{\color{red}+4}

x+0=12

x=12

 

Solve the equation: x+7=-2
Again, we begin by adding the opposite of the constant to each side:
x+7=-2

x+7+{\color{red}(-7)}=-2+{\color{red}(-7)}

Note that adding a negative can be thought of as subtracting:
x+7-7=-2-7

x=-9

We might also have use of the distributive property before utilizing our addition property.

Solve: 3t-2(t+4)=7
3t-2(t+4)=7

3t-2t-8=7

t-8=7

t-8{\color{red}+8}=7{\color{red}+8}

t=15

We might also need to use the addition property more than once in an equation.

Solve: 4y+15=3y+12
We start by subtracting 3y from each side so that we have our variable terms on one side of the equation:

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Then subtract the 15 from each side:

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Solve:  5(p-5)+8=2p+2(p-3)
Distribute and simplify first, then use the addition property twice:

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