Isolating Variables

Solving Linear Equations and Inequalities

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3 Tutorials that teach Isolating Variables

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Anthony Varela

Colleen Atakpu

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Tutorial

what's covered

Process of Solving an Equation
Review of Inverse Operations
Applying Inverse Operations
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:

		Solve for x
		Subtract 2 from both sides

$fraction numerator 3 x over denominator 3 end fraction equals 3 over 3$		Divide both sides by 3
		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:

		Solve for x
		Add 8 to both sides

$28 over 2 equals fraction numerator 2 x over denominator 2 end fraction$		Divide both sides by 2
		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:

		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
$fraction numerator 5 open parentheses 2 x minus 6 close parentheses over denominator 5 end fraction equals 7 over 5$		Divide both sides by 5
		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
		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

$fraction numerator short dash 7 over denominator 3 end fraction equals fraction numerator 3 x over denominator 3 end fraction$		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.

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