Solving Equations Using Both Addition and Multiplication Principles Video

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  • 0:03 The Addition Principle
  • 2:04 The Multiplication Principle
  • 3:12 Problem One
  • 4:12 Problem Two
  • 5:09 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

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

Most times in algebra, to solve for your variable, you will need to perform more than one operation to get your answer. Watch this video lesson to learn the proper steps to take to solve these equations.

The Addition Principle

In algebra, we do a lot of manipulation to equations. But one thing always stays the same, and it is that whatever change we make, we make sure that our equation is the same. How do we do that if we are changing numbers and such? We use what is called the addition principle, which tells us if we add or subtract a number from one side of the equation, we also need to add or subtract the same number from the other side to keep the equation the same.

Think about this for a minute. We have two piles of chocolate bars that are equal to each other. What would happen if we added two more chocolate bars to just one side, say, the left side? Would the two piles still be equal to each other? No, they wouldn't. How would we keep the two sides equal? We would have to add two chocolate bars to the other side. This is what the addition principle is all about.

If we wanted to solve an equation like x + 6 = 9, we would use the addition principle to subtract 6 from the side with the variable so that our variable is by itself. Then we would subtract the same 6 from the 9 to get our answer. We subtract because our problem has our variable being added by a 6, and subtraction is the opposite operation of addition, which will help us to separate numbers from our variable. Our answer would then be x = 3.

But what if we have an equation like 3x + 6 = 9? How would we solve this one? We use the addition principle to subtract the 6 from both sides. But then we are left with a 3x = 3. Our variable is not by itself but is being multiplied by a 3. How do we manipulate the 3 so that it separates from the x? This is where we need to use another very useful principle called the multiplication principle.

The Multiplication Principle

The multiplication principle, similar to the addition principle, tells us that if we multiply or divide by a number on one side of an equation, we also need to multiply or divide by that same number on the other side to keep the equation the same. For this principle, you can think of two groups of rabbits. Right now, they are equal to each other. But what if the rabbits in one group all gave birth to three baby rabbits each? Would the two groups be equal to each other? No, they wouldn't. The other group would also have to give birth to the same number of baby rabbits for the two groups to be the same.

How do we use this multiplication principle? The same way we did with the addition principle. If we see our variable being multiplied or divided by a certain number, we perform the opposite operation to get our variable by itself.

So, to continue solving our problem from where we left off, 3x = 3, we will divide both sides of our equation by 3. If we do this, we will get x = 1 for our answer.

Problem 1

Let's look at another problem to see how it all works together.

5x - 10 = 0

We first look to see what is going on with the variable. We see the variable on the left side being multiplied by a 5 and then subtracted by a 10. What can we tackle first, the 10 or the 5? Well, which opposite operation is simpler to perform? Addition or division? It's addition, so I will add 10 to both sides first.

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