1. Solving Equations with Variables on Both Sides
Yvonne Smith
IDT
Begin

2. Introduction
Solving equations with variables on both sides is very similar to solving equations with a variable on one side only. The same rules apply when working out the equation. Your ability to solve such equations is very important to your continued success at learning algebra. During this tutorial, you will learn how to solve equations with variables on both sides of the equation. You will also practice solving equations and take an assessment at the end in order to determine how successful you are at this skill. Upon the completion of this tutorial, you should be able to solve any equation that has variables on both sides. Good Luck and Let’s Get Started!
Throughout this tutorial you will see this navigation bar. Please use it to find your way around.

3. Before we get started, please click on the picture on the right and watch a short video on solving equations with variables on both sides. This video will help you gain a good understanding of this concept before we move forward. Now that you have watched the video, I hope you have a basic understanding of what you will be learning during this tutorial. Just as you learned when solving equations with a variable on one side only, when solving equations with variables on both sides, there are rules that must be followed. When a variable exists on both sides of the equation, the first thing you must do is get the variable onto one side of the equation. Next, you will need to solve the equation the same way you learned to solve equations with variables on one side of the equation. Here are the rules when solving equations with variables on both sides: 1) Combine like terms on each side of the equation 2) Isolate the terms that contain the variable you wish to solve for 3) Isolate the variable you wish to solve for 4) Substitute your answer into the original equation and check that it works Let’s look at some examples!
Video Tutorial

4. Take a look at this equation: 4n + 2 = 7n + 5 To solve this equation, the variable (n) must be isolated onto one side. You can do this by either moving the 4n to the right of the equation, or by moving the 7n to the left of the equation. To move either variable, you must add or subtract the variable from both sides of the equation. For example, let’s say that you decided to move the 4n to the right of the equation. In order to move the 4n, you must complete the opposite mathematical operation for 4n. 4n is a positive number, so you must subtract 4n from both sides of the equation in order to get rid of the 4n on the left. Let’s take a look: n +2 = 7n + 5 First, insert - 4n on both sides 4n – 4n +2 = 7n -4n +5 Simplify = 3n + 5 Next, you need to solve for the variable 3n, but you must isolate it first. 3n can be isolated by removing the 5 from the right of the equation. To get rid of the 5, you must subtract 5 from both sides of the equation. Insert – 5 on both sides of the equation. Your new problem would look like this: – 5 = 3n + 5 – 5 Simplify = 3n Now, solve for n by dividing both sides by 3 n = -1

5. LET ME SHOW YOU ANOTHER EXAMPLE!
Solve: 12x + 21 = 9x
First, you must isolate the x’s all on the same side. To do this, you must take the 12x AWAY from the side with the 21 by subtracting it from both sides. Take a look below:
12x + 21 = 9x
- 12x x
21 = - 3x
Now you have a one step equation to solve: divide both sides by – 3:
-7 = x
HERE IS ONE MORE EXAMPLE!
Solve: 4n = -28n – 3
First, isolate the variable on the by adding 28n to both sides
4n = - 28n – 32
+28n n
32n = - 32
Now, divide both sides by 32 in order to isolate the variable (n)
32n =
n = - 1

6. HERE IS ONE THAT IS A LITTLE BIT HARDER!
Solve: 8a - 4(-5a - 2) = 12a
First, you must get rid of the parentheses by multiplying -4 by -5a and -2. Here is the new problem:
8a + 20a + 8 = 12a
Next, combine the like terms on each side of the equation. This means to add 8a and 20a together
28a + 8 = 12a
Now , our difficult equation looks just like the other ones that we have solved!!
Next, isolate the variable (a) on one side of the equation. You can do this by subtracting the 28a which is on the left side of the equation from both sides
28a + 8 = 12a
-28a + 28a +8 = 12a – 28a
Simplify = - 16a
Solve, divide by = -16
- ½ = a

7. Now you try. Write each problem on your own paper and solve for (x)
Now you try! Write each problem on your own paper and solve for (x). When you finish,
Click on one of the answers below each problem to see if you solved the problem correctly.
11x + 7 = 3x – 9
20x – 5 – 5x = 11x + 49
Click on an answer:
Click on an answer: 2
- 1/2 -2
13 2/4
-13
13 1/2
3x – 23 = 54 – 4x
13 – 8(x – 2) = 7(x + 4) + 46
Click on an answer:
Click on an answer:
77/7 11 10 -3
45/15 3
2 + 6x – 0.2 = 5x + 2.1
Click on an answer:
0.3 3
1.8
Remember, whatever you do to one side of the equation must be done to the other side as well!

8. This is not the correct answer.
Please check your positive/negative signs
Please check your division, multiplication, addition and/or subtraction
Did you reduce to the lowest terms?
Did you combine like terms?
Please click the Practice Page tab and try again.

9. Your answer was correct! Your work might look something like this:
11x + 7 = 3x – 9
11x = 3x – 16
3x x
8x = - 16 8
x = -2
Please click the Practice Page tab and complete the next problem

10. Your work might look something like this:
Awesome Job!
Your work might look something like this:
3x – 23 = 54 – 4x
3x = 77 – 4x
+4x x
7x = 77
x = 11
Please click the Practice Page tab and complete the next problem.

11. Your work might look something like this:
Great Job!
Your work might look something like this:
2 + 6x – 0.2 = 5x + 2.1
x = 5x + 2.1
6x = 5x + 0.3
- 5x x
x = 0.3
Please click the Practice Page tab and complete the next problem.

12. Your work might look something like this:
Excellent Work!
Your work might look something like this:
20x – 5 – 5x = 11x + 49
15x – = 11x + 49
15x = 11x + 54
11x x
4x =
x = / = 13 ½
Please click the Practice Page tab and complete the next problem.

13. Your work might look something like this:
Great Job!
Your work might look something like this:
13 – 8(x – 2) = 7(x + 4) + 46
13 – 8x + 16 = 7x
8x = 7x + 74
8x = 7x + 45
7x x
-15x =
x =
You have completed the practice portion of this tutorial.
Please click the Next Page tab

14. You may review this tutorial as many times as you need to.
If you had difficulty with the practice problems, please click on the Home tab and go back and review the tutorial from the beginning.
You may review this tutorial as many times as you need to.
If you do not need further review, please go on to the Next Page.

15. If you are comfortable with the results of your practice session,
CONGRATULATIONS!
If you are comfortable with the results of your practice session,
please click the Assessment Tab below and take the test.
Your score will be recorded and saved under my profile online.
You may only take the assessment ONCE.
Good Luck!

16. References