Solving Equations Using the Addition Principle

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Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In algebra, we have certain principles that we use to help us solve equations quickly and easily. The addition principle is one of them. Watch this video lesson to learn how you can apply it to solve your equations.

The Addition Principle

The addition principle tells you that if you add or subtract the same thing to both sides of an equation, then your equation will remain the same. You can picture the addition principle by thinking of two equal piles of stuffed animals, one on the left and one on the right with an equals sign in between. What if you added some more stuffed animals to the left side? How would you keep the two piles equal to each other?

You would have to add the same amount of stuffed animals to the right side. I mentioned sides, but the principle applies no matter which side you start with. The same goes if you took away some stuffed animals from the right side. To keep the two piles equal, you would have to take away the same amount of stuff from the left side.

Why is this useful? Well, when you want to solve an equation for a particular variable, you want to isolate your variable, or get it by itself. See, sometimes, you will see your variable being added or subtracted by a number. To solve and get your variable by itself, you would need to change this number to 0 so that it is no longer adding or subtracting from our variable. In order to do this, you sometimes have to add or subtract a value from the side with the variable.

But if you only did it to one side of the equation, your answer would be wrong. So, the addition principle tells you that you have to add or subtract the same thing to the other side of the equation as well so that your equation remains the same and your answer is correct. For example, for the equation x + 1 = 3, if we subtracted the 1 from the left side only to get x by itself, I would get x = 3.

But is that a correct answer? If I plug in 3 for x in my equation, I would get 3 + 1 equaling 4. Does 4 equal 3? No! So, I can't add or subtract from just one side; I have to do it to both. Think of your piles.

An Example

Let's solve x + 1 = 3 correctly now. We see that the side with our variable also has a plus 1. How do we make this plus 1 disappear or otherwise become 0? What can I add or subtract to this plus 1 to make it 0? I need to subtract a 1. The addition principle tells me that if I subtract a 1 from one side, I also have to subtract it from the other side. Let me see what I get when I do this.

x + 1 - 1 = 3 - 1
x = 2

Ah, I get an answer of 2. Is this correct? Let me plug it back into my equation to see. 2 + 1 = 3 becomes 3 = 3. Does 3 equal 3? I think so. I am done. I have applied the addition principle, and my answer is correct!

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