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!

Practice Questions

Question #1:

 
Solve the equation: \(4x+9=25+2x\).

\(x=9\)
\(x=15\)
\(x=8\)
\(x=17\)
Answer:

To get the variable terms on the left side, subtract \(2x\) from both sides of the equation.

\(4x+9-2x=25+2x-2x\)
\(2x+9=25\)

Now, subtract 9 from both sides.

\(2x+9-9=25-9\)
\(2x=16\)

Finally, divide both sides by 2.

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

\(x=8\)

Question #2:

 
Solve the equation: \(-5x-16=-7x+4\).

\(x=-14\)
\(x=12\)
\(x=10\)
\(x=-10\)
Answer:

To get the variable terms on the left side, add \(7x\) to both sides of the equation.

\(-5x-16+7x=-7x+4+7x\)
\(2x-16=4\)

Now, add 16 to both sides.

\(2x-16+16=4+16\)
\(2x=20\)

Then, divide both sides by 2.

\(\frac{2x}{2}=\frac{20}{2}\)

\(x=10\)

Question #3:

 
Solve the equation: \(6x-11=2x+27\).

\(x=-\frac{19}{2}\)
\(x=\frac{19}{2}\)
\(x=\frac{7}{2}\)
\(x=-\frac{7}{2}\)
Answer:

To get the variable terms on the left side, subtract \(2x\) from both sides of the equation.

\(6x-11-2x=2x+27-2x\)
\(4x-11=27\)

Now, add 11 to both sides.

\(4x-11+11=27+11\)
\(4x=38\)

Then, divide both sides by 4.

\(\frac{4x}{4}=\frac{38}{4}\)

\(x=\frac{38}{4}\)

Reduce the fraction by 2, to write the answer in simplest form.

\(x=\frac{38\div2}{4\div2}=\frac{19}{2}\)

Question #4:

 
You charge homeowners in your neighborhood a flat fee of $20 plus $10 per hour to rake leaves from their yards. Your friend charges a flat fee of $35 plus $7 per hour for the same service. How many hours will each of you need to work to make the same amount of money raking leaves in your neighborhood?

5 hours

3 hours

10 hours

8 hours

Answer:

Let \(x\) be the number of hours you both work. You have a flat fee of $20 and are charging $10 per hour, the amount of money you make for working \(x\) number of hours is \(20+10x\). Your friend has a flat fee of $35 and charges $7 per hour, so the amount of money your friend makes for working \(x\) number of hours is \(35+7x\). To determine how many hours you each need to work to make the same amount of money, we need to set the two expressions equal to each other.

\(20+10x=35+7x\)

To get the variable terms on the left side, subtract \(7x\) from both sides of the equation.

\(20+10x-7x=35+7x-7x\)
\(20+3x=35\)

Now, subtract 20 from both sides.

\(20+3x-20=35-20\)
\(3x=15\)

Finally, divide both sides by 3.

\(\frac{3x}{3}=\frac{15}{3}\)

\(x=5\)

Question #5:

 
You and your friend are on the school track team. You each spend a week running alone, then you and your friend run at the same time. During the week alone, you ran 16 miles and your friend ran 23 miles. While running at the same time, you run 4 miles each day and your friend runs 3 miles each day. How many days after you starting running together will you and your friend have run the same number of miles?

3 days

4 days

9 days

7 days

Answer:

Let \(x\) be the number days you each run after the week of running alone. Since you run 4 miles each day, you run \(4x\) miles in \(x\) days. Similarly, since your friend runs 3 miles each day, she runs \(3x\) miles in \(x\) days. During the week you each ran alone, you ran 16 miles, and your friend ran 23 miles. So, after the first week of running alone, you will have run \(16+4x\) miles, and your friend will have run \(23+3x\) miles.

Now, set the expressions equal to each other to find how many days it will take for you and your friend to run the same number of miles.

\(16+4x=23+3x\)

To get the variable terms on the left side, subtract \(3x\) from both sides of the equation.

\(16+4x-3x=23+3x-3x\)
\(16+x=23\)

Now, subtract 16 from both sides.

\(16+x-16=23-16\)
\(x=7\)

Therefore, 7 days after you and your friend started running together, you both will have run the same number of miles.

 

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by Mometrix Test Preparation | This Page Last Updated: April 27, 2022