Sometimes we’ll have to solve equations where there are variables on both sides of the equations. In these types of situations, our first goal is to get all of the terms that have the variable on the same side of the equation. Then we combine like terms, and proceed normally to solve the equation. Let’s look at an example.
3x + 4 = 5x – 14
To move all of the variables to the left side of the equation, we would need to get rid of the 5x on the right side of the equation. To do so, we’ll need to subtract 5x from the right side of the equation. To keep the equation balanced, we then have to subtract 5x from the left side of the equation.
3x + 4 = 5x – 14
3x + 4 – 5x = 5x – 14 – 5x
Remember that we can change subtraction to addition by changing the sign of the second term. Here we’re going to change all of our subtraction into addition so that we can use the commutative property of addition to move our terms around.
3x + 4 – 5x = 5x – 14 – 5x
3x + 4 + -5x = 5x + -14 + -5x
3x + -5x + 4 = 5x + -5x + -14
-2x + 4 = -14
Now we have an equation that we already know how to solve. We continue:
-2x + 4 = -14
-2x + 4 – 4 = -14 – 4
-2x = -18
-2x/(-2) = -18/(-2)
x = 9
And finally, as always, we check our answer by plugging it into the original equation to see if everything checks out.
3x + 4 = 5x – 14
3(9) + 4 = 5(9) – 14
27 + 4 = 45 – 14
31 = 31
And so we’re good. One more example:
7x – 8 = 3x + 20
Once again, we need to get rid of the 3x, so we’re going to subtract 3x from both sides.
7x – 8 – 3x = 3x + 20 – 3x
Now we change everything to addition so we can move terms around.
7x + -8 + -3x = 3x + 20 + -3x
Next, we move our terms around so that we can simplify like terms.
7x + -3x + -8 = 3x + -3x + 20
4x – 8 = 20
And from here we proceed normally.
4x – 8 = 20
4x – 8 + 8 = 20 + 8
4x = 28
4x/4 = 28/4
x = 7
Finally, we check our answer.
7x – 8 = 3x + 20
7(7) – 8 = 3(7) + 20
49 – 8 = 21 + 20
41 = 41
And now we know our answer was the right one.