In order to understand the equations in this lesson, it's important that you understand how to solve one-step equations and two-step equations. Equations with variables on both sides involve a few more steps, so we want to make sure you understand the simpler equations first before moving forward. Let's say you have the equation below:
This equation means that if you multiply a "mystery number" x by 5 and then subtract 1, the answer will be 19. To solve for x, you need to work your way backwards to get x by itself. Since the subtraction was done last, that's the first thing that needs to be undone. How do you cancel out subtracting 1? You need to use the inverse operation and add 1 to both sides. Remember to always to the same thing on BOTH sides of the equation to keep things balanced.
Now we're left with the equation 5x = 20. This means that when you multiply x by 5, you get 20. How do you undo multiplying by 5? You use the inverse operation and divide both sides by 5.
Once you have your answer, it's a good idea to check it to make sure it's correct. To check our answer, we can just plug in 4 for x in the original equation and make sure it comes out to 19.
If you multiply 4 by 5 and then subtract 1, it does come out to 19. This means we have the correct answer.
Solving Equations with Variables on Both Sides
Sometimes you'll get an equation that's a little more complicated than the one above. If you're in an algebra class, you'll likely run into an equation like this:
What's the big difference between this equation and the one we solved above? This equation has variables on both sides of the equals signs. Remember, you can visualize an equation using a balance scale. We could use the picture below to represent the equation:
On the left, there are 5 boxes with x in them. On the right, there are 3 boxes with x in them. Remember, if we do anything to the equation we have to make sure that we do the same thing to both sides to keep it balanced. Even though we don't know what x is, we know that each box containing x weighs the same amount. If we take 3 of the boxes off each side, it will still be balanced because each box weighs the same.
Anytime you have an equation with variables on both sides, it's a good idea to try to cancel them out on one side. The goal is to get all the variables on the same side - this will give you a simpler equation to solve. You can add or subtract terms with variables in them, just like you add or subtract numbers on each side.
Once you've moved all the variables to one side of the equation, you're left with an equation that's easier to solve. The equation 2x + 1 = 5 means that when you multiply x by 2 and then add 1, you end up with 5. To solve this for x, we need to work our way backwards to get x by itself. This is a two-step equation that's probably similar to one you've seen before. If not, check out the lesson on solving two-step equations for more help. To solve for x, we need to subtract 1 from both sides first, then divide both sides by 2.
The more complicated your equation is, the more important it is to check your answer. This equation has an x on both sides of the equals sign, so we need to plug in 2 for x on BOTH sides to check our answer. If both sides come out to the same number, that means our answer is correct.
Notice that to check the answer, we had to plug in 2 for x on both sides. We simplified both sides after plugging in 2 and both sides came out to the same number. This means we have the right answer. If you go to check your answer and the two sides come out to two different numbers, that means you have a mistake somewhere that you need to go back and try to fix.
Let's try another example.
Example 2
When you have variables on both sides, the goal is to get all the variables on the same side first. You have a choice to either try to cancel out the variable term on the left or cancel the variable term on the right. In other words, you can either cancel out the -2x on the left or cancel the 3x on the right. Either choice is fine, you'll get the same answer either way. We prefer to avoid negative numbers if we can, so we're going to cancel out the -2x by adding 2x to both sides. This will give us a positive 5x on the right. Get all the variables on the same side first:
Now that all the variables are on the same side, it's a much simpler equation to solve. 12 = 5x - 8 means that if you multiply x by 5 and then subtract 8, the answer should be 12. We need to solve this by working backwards and undoing the subtraction first. We can undo subtracting 8 by adding 8 to both sides. Last, we can undo multiplying by 5 by dividing both sides by 5.
It's always a good idea to check your work whenever possible so you know that you have the right answer. To make sure the answer is 4, we need to plug 4 in for x on both sides. Then simplify both sides and make sure they both come out to the same number.
When we plugged in 4 for x, both sides came out to the same number. This means our answer is correct. If you go to check your answer and get something like 5 = 7, with two different numbers, that means you made a mistake somewhere along the way. If this happens, go back to the beginning and double check your work. It's common to make mistakes when negative numbers are involved. Double check your work to make sure you didn't drop a negative sign along the way. It may also be helpful to double check your steps with a calculator to make sure you didn't make simple addition or subtraction errors.