Solving Equations with Variables on Both Sides Page 2 of 4
Example 1 (continued)
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.
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.
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.
Remember to work your way backwards to get x by itself. Undo addition and subtraction before you undo multiplication and division.
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.