When solving an equation for a variable, our main goal is to isolate a variable. In other words, we want to get the variable by itself on one side of the equation, with all other expressions on the other side of the equals sign. In this process, we must always remember that if we perform an operation on one side of the equal sing, we must do the same on the other side of the equal sign. Let's look at an example:

Solve for x  

Subtract 2 from both sides  




Divide both sides by 3  

Our Solution 
When isolating a variable, we need to keep the following in mind:
Operation  Inverse Operation 

Addition  Subtraction 
Subtraction  Addition 
Multiplication  Division 
Division  Multiplication 
Powers  Roots 
Roots  Powers 
A good rule of thumb is to isolate the outermost operations surrounding the variable first, working our way inwards until we isolate the variable. Let's look at an example:

Solve for x  

Add 8 to both sides  




Divide both sides by 2  

Our Solution 
Sometimes when we try to isolate a variable, it may be better to simplify the equation before we perform any inverse operations. This is illustrated below:

Solve for x 
There are two ways we can go about solving this equation. First, we can distribute 5 into the 2x and –6, and then isolate the variable, or we can divide both sides of the equation by 5 first, and then solve for x. Either method is value, and you are free to use either when trying to isolate the variable. Let's take a look at how we can use both methods to solve the equation above:
By distribution:

Solve for x  

Distribute 5 into  

Add 30 to both sides  

Divide both sides by 10  

Our Solution 
Dividing 5 first:

Solve for x  

Divide both sides by 5  

Express 7/5 as a decimal  

Divide both sides by 2  

Our Solution 
Let's look at another example where combining like terms before attempting to isolate the variable can be helpful:

Solve for x  

Move the x terms to one side  



Move constant term to one side  



Divide both sides by 3  

Our Solution 