Solving equations with variables on both sides may seem daunting at first, but once you learn how to isolate the variable by moving it to one side of the equation, the problem will become much easier to handle. Here are some examples for you to review to practice this technique.
Steps
Method 1 of 5: Solve with a Variable on Both Sides
Step 1. Examine the equation
When it comes to an equation that has only one variable on both sides, the goal is to put the variable on one side to solve it. Check the example to determine the best way to proceed.
20 - 4 x = 6 x
Step 2. Isolate the variable from one side
You can isolate the variable by adding or subtracting the variable with its corresponding coefficient from either side of the equation. You need to add or subtract for 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 - 4 x + 4 x = 6 x + 4 x
- 20 = 10 x
Step 3. Simplify both sides through parting
When a coefficient stays in front of the variable, remove it, dividing both sides by that number. You need to divide both sides by that value in order to keep the equation balanced. By performing this step, you should isolate the variable, allowing the equation to be solved.
- 20/10 = 10 x / 10
- 2 = x
Step 4. Test
Verify that your answer is correct by inserting the found value in place of the variable in the equation each time it appears. If both sides of the equation are equal, congratulations - you have solved the equation correctly!
- 20 – 4 (2) = 6 (2)
- 20 – 8 = 12
- 12 = 12
Method 2 of 5: Perform an Example Problem
Step 1. Examine the equation
When it comes to an equation that has only one variable on both sides, the goal is to have the variable on one side only to solve it. For some equations, additional steps need to be developed before the variable can be brought to one side.
5 (x + 4) = 6 x - 5
Step 2. Use the distributive property if necessary
When dealing with an equation that has an expression in parentheses, such as 5 (x + 4), you need to distribute the value outside the parentheses for the numbers inside using multiplication. This is a necessary step to proceed.
- 5 x + (5) 4 = 6 x - 5
- 5 x + 20 = 6 x - 5
Step 3. Isolate the variable from one side
After removing the parentheses from the equation, take the standard measures required to isolate the variable from a single side of the equation. Add or subtract the variable, with its corresponding coefficient, to both sides of the equation. Both sides must be added or subtracted in order to keep the equation balanced. Choose a variable-coefficient pair already present in the equation and, when possible, choose to shift that pair which will create a positive coefficient value.
- 5 x + 20 - 5 x = 6 x - 5 - 5 x
- 20 = x - 5
Step 4. Simplify both sides by subtracting or adding
Sometimes, additional numbers will be left on the side of the equation containing the variable. Remove these numeric values by adding or subtracting them from both sides. You need to add or subtract values from both sides in order to keep a balanced equation.
- 20 + 5 = x - 5 + 5
- 25 = x
Step 5. Test
Check the solution by entering the value found in the variable, each time it appears. If both sides of the equation are equal, congratulations - you have solved the equation correctly!
- 5(25 + 4) = 6 (25) – 5
- 125 + 20 = 150 – 5
- 145 = 145
Method 3 of 5: Solve Another Example Problem
Step 1. Examine the equation
When it comes to an equation that has only one variable on both sides, the goal is to shift the variable to one side to solve it. Some equations will require additional steps before the variable can be isolated to one side.
7 + 3 x = (7 - x) / 2
Step 2. Remove any fractions
If a fraction is displayed on both sides of the equation, you must multiply both sides of the equation with the denominator in order to remove the fraction. Perform this action on both sides of the equation to keep it balanced.
- 2 (-7 + 3 x) = 2 [(7 - x) / 2]
- -14 + 6 x = 7 - x
Step 3. Isolate the variable from one side
Add or subtract the variable with its coefficient from both sides of the equation. You need to perform the same action on both sides. Choose a variable-coefficient pair that is already in use and, if possible, choose to move a pair that will create a positive coefficient in front of the variable.
- -14 + 6 x + x = 7 - x + x
- -14 + 7 x = 7
Step 4. Simplify both sides by subtracting or adding
When the additional numbers are left on the side of the equation containing the variable, remove them, adding or subtracting them from both sides. You need to add or subtract values from both sides in order to keep the equation balanced.
- -14 + 7 x +14 = 7 +14
- 7 x = 21
Step 5. Simplify both sides through the parting
When a coefficient stays in front of the variable, remove it, dividing both sides by that coefficient. You have to divide both sides by the same value. By performing this step you should isolate the variable and arrive at the solution of the equation.
- (7 x) / (7) = 21/7
- x = 3
Step 6. Test
Verify that your answer is correct by inserting the found value in place of the variable in the equation. If both sides of the equation are equal, congratulations - you have solved the equation correctly!
- -7 + 3 (3) = (7 – (3))/2
- -7 + 9 = (4)/2
- 2 = 2
Method 4 of 5: Solve with Two Variables
Step 1. Examine the equation
When you have a single equation with several variables on either side of the equal sign, you won't be able to get a complete answer. You can solve for any variable, but the solution will always contain the other.
2 x = 10 - 2 y
Step 2. Solve for x
Follow the same standard procedure you use when extracting a variable. Simplify the equation, if needed, in order to isolate that variable on one side of the equation, with no additional elements. Note that, in the following example, when we solve for x, we expect to see y in the solution.
- (2 x) / 2 = (10 - 2 y) / 2
- x = 5 - y
Step 3. Alternatively, you can solve for y
Follow the standard procedure you use when calculating a variable. Use addition, subtraction, multiplication and division, if needed, to simplify the equation, then isolate that variable on one side of the equation without any additive constants. Note that when we find y in the following example, we expect to see x in the solution.
- 2 x - 10 = 10 - 2 y -10
- 2 x - 10 = - 2 y
- (2 x - 10) / -2 = (- 2 y) / -2
- - x + 5 = y
Method 5 of 5: Solving Systems of Equations with Two Variables
Step 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 a variable is isolated from one side of one of the equations before proceeding.
- 2 x = 20 - 2 y
- y = x - 2
Step 2. Replace the equation of one variable into another equation
If you haven't already done so, isolate the variable in one of the equations. Replace the value of this variable - which at this point will be in the form of an equation - in the same variable, but in the other equation. Doing so transform the equation from two to a single variable, present on both sides.
2 x = 20 - 2 (x - 2)
Step 3. Solve for the remaining variable
Follow the usual steps required in order to isolate the variable and simplify the equation, then find the solution of the variable that remains in the equation.
- 2 x + 2 x = 20 - 2 x + 4 + 2 x
- 4 x = 20 + 4
- 4 x = 24
- 4 x / 4 = 24/4
- x = 6
Step 4. Enter this value in one of the two equations
Once you have the solution of a variable, you should substitute that solution in one of the two equations of the system to determine what the value of the second variable is. Generally, it is easier to do this with the equation where the second variable is already isolated.
- y = x - 2
- y = (6) - 2
Step 5. Find the other variable
Make all the calculations necessary to solve the second variable.
y = 4
Step 6. Test
Double-check your answer by inserting the values of the two variables into all equations. If both sides of the equal sign are equivalent, then congratulations: you have successfully found the value of both variables.
- 2 (6) = 20 – 2 (4)
- 12 = 20 – 8
- 12 = 12
