Solving Equations with Variables on Both Sides

Hello! Today we’re going to take a look at solving equations with variables on both sides. The solving process for these types of problems will look similar to more simple forms of solving equations, it will just require a few additional steps. So, let’s start by taking a look at an example.

\(5x+9=3x-7\)

To solve problems like these, you will want to get all the variable terms on one side of the equation and all the constant terms on the other side of the equation. It doesn’t matter which side either one goes on, just make sure that all the variable terms are on one side and all the constants are on the other side. So, for this, I’m going to get the \(x\)-terms on the left side of the equation. So, to do that, we’ll subtract \(3x\) from both sides.

\(5x-3x+9=3x-3x-7\)

If you wanted to subtract \(5x\) from both sides and get the \(x\)-terms on the right side of the equation, that would be totally fine as well.

\(2x+9=-7\)

So now, to get the constants on the other side, we’re going to want to subtract 9 from both sides.

\(2x+9-9=-7-9\)
 
\(2x=-16\)

And finally, to solve the equation, we need to get \(x\) by itself, so we do that by dividing both sides by 2.

\(\frac{2x}{2}=\frac{-16}{2}\)
 
\(x=-8\)

Let’s take a look at another problem.

\(-2x+13=6x-31\)

This time, we are going to solve for the variable on the right side. So, to do this, we’re going to start by adding \(2x\) to both sides of the equation — that will get our \(x\)-terms on the right side.

\(-2x+2x+13=6x+2x-31\)
 
\(13=8x-31\)

Now, we want to get our constants on the left side so we’ll add 31 to both sides.

\(13+31=8x-31+31\)
 
\(44=8x\)

And finally, to get \(x\) by itself, we divide both sides by 8.

\(\frac{44}{8}=\frac{8x}{8}\)
 
\(\frac{44}{8}=x\)

Now, this fraction isn’t in most simplest form, so the best thing to do is to simplify it. The way that we can simplify this fraction is by dividing both the numerator and denominator by 4.

\(x=\frac{44}{8}=\frac{44\div 4}{8\div 4}=\frac{11}{2}\)
 
\(x=\frac{11}{2}\)

Let’s take a look at one more problem before we go.

\(4x+17=-9x-9\)

So, first we’re going to add \(9x\) to both sides. If you wanted to subtract 4x from both sides and solve for \(x\) on the right side of the equation, that would totally be acceptable as well. The reason I chose to add \(9x\) to both sides is because I don’t particularly like working with negative numbers when I don’t have to. So I’m adding \(9x\) so that I’ll have a positive number of \(x\)‘s on the left side. So, that’s why I chose that, but if you like negatives and want to do that, you are more than welcome to.

\(4x+9x+17=-9x+9x-9\)
 
\(13x+17=-9\)

Now I’m going to subtract 17 from both sides of our equation.

\(13x+17-17=-9-17\)
 
\(13x=-26\)

So, notice we have a negative over here, so we’re dealing with negatives anyways, so it’s just up to you back at this first step if you want to have negative \(x\)-values or negative constants. Either way will get you the right answer. So to get \(x\) by itself and solve for \(x\) we’re going to divide by 13 on both sides.

\(\frac{13x}{13}=\frac{-26}{13}\)
 
\(x=-2\)

I hope that this video gave you more confidence in solving equations. Thanks for watching and happy studying!

Solving Equations with Variables on Both Sides Practice Questions