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Isolating Variables

Isolating Variables

Author: Sophia Tutorial
Description:

Identify the operations needed to isolate a variable in an equation.

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Tutorial
what's covered
  1. Process of Solving an Equation
  2. Review of Inverse Operations
  3. Applying Inverse Operations
  4. Simplifying Before Isolating a Variable

1. Process of Solving an Equation

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:

3 x plus 2 equals 5
Solve for x
negative 2 space space space space space space space minus 2
Subtract 2 from both sides
3 x equals 3

fraction numerator 3 x over denominator 3 end fraction equals 3 over 3
Divide both sides by 3
x equals 1
Our Solution

big idea
Whatever we do on one side of the equation has to be done on the other side of the equation. This is known as the Rule of Equality.


2. Review of Inverse Operations

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

3. Applying Inverse Operations

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:

20 equals 2 x minus 8
Solve for x
plus 8 space space space space space space space plus 8
Add 8 to both sides
28 equals 2 x

28 over 2 equals fraction numerator 2 x over denominator 2 end fraction
Divide both sides by 2
14 equals x
Our Solution

hint
In general, we apply the inverse operations following the reverse order of operations to isolate a variable.


4. Simplifying Before Isolating a Variable

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:

5 open parentheses 2 x minus 6 close parentheses equals 7
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:

5 open parentheses 2 x minus 6 close parentheses equals 7
Solve for x
10 x minus 30 equals 7
Distribute 5 into 2 x minus 6
10 x equals 37
Add 30 to both sides
x equals 37 over 10
Divide both sides by 10
x equals 3.7
Our Solution

Dividing 5 first:

5 open parentheses 2 x minus 6 close parentheses equals 7
Solve for x
fraction numerator 5 open parentheses 2 x minus 6 close parentheses over denominator 5 end fraction equals 7 over 5
Divide both sides by 5
2 x minus 5 equals 1.4
Express 7/5 as a decimal
fraction numerator 2 x over denominator 2 end fraction equals fraction numerator 7.4 over denominator 2 end fraction
Divide both sides by 2
x equals 3.7
Our Solution

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

2 x minus 4 equals 5 x plus 3
Solve for x
negative 2 x space space space space space space space space space space space minus 2 x
Move the x terms to one side
short dash 4 equals 3 x plus 3
open parentheses 5 x minus 2 x equals 3 x close parentheses
space space space space space space space space minus 3 space space space space space space space space space space space space space space space space space minus 3
Move constant term to one side
short dash 7 equals 3 x
open parentheses short dash 4 minus 3 equals short dash 7 close parentheses
fraction numerator short dash 7 over denominator 3 end fraction equals fraction numerator 3 x over denominator 3 end fraction
Divide both sides by 3
fraction numerator short dash 7 over denominator 3 end fraction equals x
Our Solution

hint
When trying to isolate a variable, it is always a good idea to simplify the equation as much as possible before starting to isolate the variable with inverse operations. This usually means that we should combine like terms whenever possible.
summary
The process of solving an equation involves isolating the variable you want to solve for. When isolating a variable, it is helpful to have a review of inverse operations: addition and subtraction are inverse, multiplication and division are inverse and powers and roots are inverse. Keep in mind when applying inverse operation that this will cancel the operations around the variable. Also, in using the inverse operations, use the order of operations in reverse order. Finally, simplifying before isolating a variable, such as distributing or combining like-terms, can be helpful.