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.
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.
5x - 10 + 10 = 0 + 10, which simplifies to 5x = 10.
Okay, now I am left with my variable being multiplied by 5. Now I can apply the multiplication principle and divide both sides by 5.
5x/5 = 10/5, which simplifies to x = 2. I am done and my answer is 2.
Now, it's your turn. Try to solve this problem with me.
x/2 - 3 = 3
We see a division by 2 and a subtraction by 3. The 3 is the simpler operation. To remove the 3 from the variable, we need to perform the opposite operation, which is addition. So, we add 3 to both sides using the addition principle.
x/2 - 3 + 3 = 3 + 3, which simplifies to x/2 = 6.
Now, we have division by 2. To remove the 2, we perform the opposite operation and multiply. We use the multiplication principle to multiply the 2 on both sides.
(x/2) * 2 = 6 * 2, which simplifies to x = 12.
Our variable is by itself and we have our answer of 12.
What have we learned? We've learned that in algebra, when we want to keep an equation the same while manipulating it, we use the help of two principles. They are the addition principle and the multiplication principle. The addition principle tells us that 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.
The multiplication 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. To solve an equation where we need to perform both types of operations, we use both of these principles to help us solve.
We always tackle any addition or subtraction first. Then, we tackle any multiplication or division last. To remove numbers from our variable, we always perform the opposite operation. If we see addition, we subtract. If we see subtraction, we add. If we see division, we multiply. If we see multiplication, we divide.
Some expected outcomes of this lesson include the ability to:
- Understand the functions of the addition and multiplication principles
- Work through related practice problems
Practice Solving Equations
In the following practice problems, students will solve equations using the addition and multiplication principles. They will also try to find a common mistake in a problem and then explain what went wrong with that solution.
1. State which principle(s) need to be used and then solve the equation: x + 7 = 12.
2. State which principle(s) need to be used and then solve the equation: 4x = 32.
3. State which principle(s) need to be used and then solve the equation: 3x - 10 = 11.
4. State which principle(s) need to be used and then solve the equation: x / 5 + 9 = 54.
5. When solving the problem 3x - 6 = 12, Sam did the following steps:
- Subtract 6 from both sides to get 3x = 6.
- Divide both sides by 3 to get x = 2.
Sam's answer is incorrect. Which step did Sam do incorrectly? What is the correct approach to the problem? What is the correct answer?
1. The addition principle needs to be used. As 7 has been added to the variable, we need to perform the opposite operation (subtract 7) on both sides of the equation. This gives us x = 12 - 7, which is x = 5.
2. The multiplication principle needs to be used. The variable has been multiplied by 4, so we need to perform the opposite operation (divide by 4) on both sides of the equation. This gives us x = 32 / 4, which is x = 8.
3. First, we need to use the addition principle. Add 10 to both sides of the equation to get 3x = 21. Then use the multiplication principle and divide both sides of the equation by 3 to get x = 7.
4. First, we need to use the addition principle. Subtract 9 from both sides of the equation to get x / 5 = 45. Then use the multiplication principle and multiply both sides of the equation by 5 to get x = 225.
5. Sam's first step was incorrect - the addition principle was the correct principle to use, but the original problem involved subtracting 6, so we need to perform the opposite operation and add 6 to both sides of the equation. This would result in 3x = 18. Then divide both sides of the equation by 3 to get x = 6. It's very important to remember to perform the opposite operation when using the addition and multiplication principles to solve equations. When Sam tried to subtract 6 from both sides of the equation, it was actually done incorrectly, since it would result in 3x - 12 = 6 not 3x = 6. Sam incorrectly thought that subtracting 6 would get rid of the " - 6" on the left side of the equation.