How to Solve Linear Functions

Instructor: Maria Airth

Maria has a Doctorate of Education and over 20 years of experience teaching psychology and math related courses at the university level.

Solving linear functions is a common and important part of mathematics. In this lesson, you will find step by step instructions for solving linear functions. You will also see how to check your work and see examples of alternate versions of linear function solutions.

Solving Linear Functions

A linear function is a function with the form f(x)=ax + b. It looks like a regular linear equation, but instead of using y, the linear function notation is f(x) (spoken as 'f of x'). To solve a linear function, you would be given the value of f(x) and be asked to find x.

Let's go through the steps with the help of an example:

  • f(x)=3x-1, solve for f(x)=8

Step 1

Substitute the value of f(x) into the problem.

  • 8=3x-1

Step 2

Isolate the variable. Add 1 to both sides to isolate the variable term by using the opposite operation to move the constant term across the equal sign.

  • 8 +1 = 3x - 1 +1; 9 = 3x

Step 3

Continue to isolate the variable. Divide by 3 on both sides to isolate the variable itself.

  • 9/3 = 3x/3

Step 4


  • x=3


Because this is a linear function, it is appropriate to format the final answer in function form.

For the function f(x) = 3x-1; f(3) = 8.

How to Check Your Answer And More Examples

Check Your Answer

To check your answer, just substitute both the f(x) value and the x value into your original function. When simplified, both sides of the equal sign should match.

f(x) = 3x - 1; f(3)=8

8 = 3(3) - 1

8 = 9 - 1

8 = 8 Perfect!

More Examples

Given X Value

If you are given the value of x, you can solve for f(x) in the same way. Just follow the steps to isolate the unknown (f(x)).

If f(x) = 5x - 1, what is f(2)? Here, we are told that x=2 by the substitution of 2 in the function format f(x).

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