5 Methods to Solving Equations with Variables on Both Sides

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NOTE: Compare the problem(s) you are doing; to make sure you are looking at the method you need to use.

 

Method 1

1

Examine the equation. When dealing with an equation that has a single variable on both sides, the objective is to get the variable on one side in order to solve it. Check the occasion to determine the best way to go about doing so.

• 20 - 4x = 6x   

2 
Isolate the variable to one side. You can isolate the variable by adding or subtracting the variable with its corresponding coefficient to both sides of the equation. You must add or subtract to both sides in order to keep the equation balanced. Choose a variable-coefficient pair already in the equation, and when possible, choose to move a pair that will create a positive value for the coefficient in front of the variable.

• 20 – 4x + 4x = 6x + 4x
• 20 = 10x 

3
Simplify both sides through division. When a coefficient remains in front of the variable, remove it by dividing both sides by that coefficient. You must divide both sides by that value in order to keep the equation balanced. Performing this step should isolate the variable, thereby allowing you to solve the equation.

• 20 / 10 = 10x / 10
• 2 = x 

4 
Check the equation. Verify that your answer is correct by plugging the value you found for the variable back into the equation, using it to stand in for the variable whenever that variable appeared. If both sides of the equation are equal, congratulations – you solved the equation correctly.

• 20 – 4(2) = 6(2)
• 20 – 8 = 12
• 12 = 12

Method 2

 

1

Examine the equation. When dealing with an equation that has a single variable on both sides, the objective is to get the variable on one side in order to solve it. For some equations, you will need to take additional steps before you can bring the variable over to one side.

• 5(x + 4) = 6x - 5

2

Distribute as needed. When dealing with an equation that has a parenthetical expression, such as 5(x + 4), you must distribute the value outside of the parentheses to the values inside using multiplication. This is a necessary step to take before proceeding.

• 5x + (5)4 = 6x – 5
• 5x + 20 = 6x – 5

3

Isolate the variable to one side. After removing the parentheses from the equation, take the standard steps required to isolate the variable to a single side of the equation. Add or subtract the variable with its corresponding coefficient to both sides of the equation. You must add or subtract to both sides in order to keep the equation balanced. Choose a variable-coefficient pair already in the equation, and when possible, choose to move a pair that will create a positive value for the coefficient in front of the variable.

• 5x + 20 -5x = 6x – 5 -5x
• 20 = x – 5

4

Simplify both sides through subtraction or addition. Sometimes, additional numbers will be left on the side of the equation containing the variable. Remove these numerical values by adding or subtracting them to both sides. You must add or subtract the values to both sides in order to maintain a balanced equation.

• 20 +5' = x – 5 +5
• 25 = x

5

Check the equation. Double-check your solution by plugging the value back in, using it to stand in for the variable whenever that variable appeared. If both sides of the equation are equal, congratulations – you solved the equation correctly.

• 5(25 + 4) = 6(25) – 5
• 125 + 20 = 150 – 5
• 145 = 145

 Method 3

1

Examine the equation. When dealing with an equation that has a single variable on both sides, the objective is to get the variable on one side in order to solve it. Some equations will require additional steps before the variable can be isolated to one side.

• -7 + 3x = (7 - x)/2

 

2

Remove any fractions. If a fraction appears on either side of the equation, you should multiply both sides of the equation with the denominator in order to remove the fraction. Perform this action to both sides of the equation to keep it balanced.

• 2(-7 + 3x) = 2[(7 – x)/2]
• -14 + 6x = 7 - x

 

3

Isolate the variable to one side. Add or subtract the variable and any corresponding coefficient to both sides of the equation. You must perform the same action both sides of the equation. Choose a variable-coefficient pair already in use, and when possible, choose to move a pair that will create a positive value for the coefficient in front of the variable.

• -14 + 6x +x = 7 – x +x
• -14 + 7x = 7

4

Simplify both sides through subtraction or addition. When additional numbers are left on the side of the equation containing the variable, remove these numerical values by adding or subtracting them to both sides. You must add or subtract the values to both sides in order to maintain a balanced equation.

• -14 + 7x +14 = 7 +14
• 7x = 21

5

Simplify both sides through division. When a coefficient remains in front of the variable, remove it by dividing both sides by that coefficient. You must divide both sides by the same value. Performing this step should isolate the variable, thereby allowing you to solve the equation.

• (7x)/(7)= 21/7
• x = 3

6

Check the equation. Check your work by substituting your solution for the variable in the original equation. If both sides of the equation are equal, congratulations — you've solved the equation.

• -7 + 3(3) = (7 – (3))/2
• -7 + 9 = (4)/2
• 2 = 2

Method 4

 

1

Examine the equation. When you have a single equation with different variables on either side of the equal sign, you will not be able to reach a complete answer. You can solve for either variable, but your solution will contain the other variable.

• 2x = 10 - 2y

2

Solve for x. Follow the standard procedure you would use when solving a variable. Simplify the equation as needed in order to isolate that variable on one side of the equation, without any additional elements. Note that when solving for x in the below example, expect to see y in your solution.

• (2x)/2 = (10 – 2y)/2
• x = 5 - y 

3 

Alternatively, solve for y. Follow the standard procedure you would use when solving a variable. Use addition, subtraction, division, and multiplication as needed to simplify the equation, thereby isolating that variable on one side of the equation without any additional elements. Note that when solving for y in the below example, expect to see x in your solution.

• 2x - 10 = 10 - 2y -10
• 2x – 10 = - 2y
• (2x – 10)/-2 = (- 2y)/-2
• -4x + 5 = y

Method 5

1

Examine the set of equations. If you have a set or system of equations with different variables on opposite sides of the equal sign, you can solve for both variables. Make sure that one variable is isolated to one side of one of the equations before proceeding.

• 2x = 20 - 2y
• y = x - 2

2

Plug the variable equation from one equation into the other. Isolate the variable in one of the equations if it has not already been done. Substitute the value of that variable—which will be in equation form at this point—for the same variable in the other equation. Doing so turns the equation into a single variable equation with a variable on both sides.

• 2x = 20 - 2(x - 2)

3

Solve for the remaining variable. Follow the usual steps needed in order to isolate the variable and simplify the equation, thereby finding your solution for the variable that remains in the equation.

• 2x + 2x = 20 - 2x + 4 + 2x
• 4x = 20 + 4
• 4x = 24
• 4x/4 = 24/4
• x = 6

4

Plug this value into either equation. Once you have the solution for one variable, you should plug that solution into either equation in the system to determine what the value for the second variable is. Generally, it is easiest to do this to the equation in which the second variable is already isolated.

• y = x – 2
• y = (6) – 2

5

Solve for the other variable. Do the necessary math to solve for the second variable.

• y = 4

6

Check your answer. Double-check your answer by plugging both values for both variables into one or both equations where appropriate. If both sides of the equal sign are balanced and equal, then congratulations — you've successfully found the value of both variables.

• 2(6) = 20 – 2(4)
• 12 = 20 – 8
• 12 = 12

Step-by-Step Solving Variables on both sides of the = video