# Solving 3×3 Systems of Equations using Matrices

These lessons, with videos, examples, solutions, worksheets and activities, help Algebra students learn how to solve 3×3 systems of equations using the inverse of matrices. We also have a matrix calculator that you can use to calculate the inverse of a 3×3 matrix.

How to solve a system of three linear equations with three unknowns using a matrix equation?

Example:
Solve the system using a matrix equation. (Use a calculator)
x + 2y - z = 7
2x - 3y - 4z = -3
x + y + z = 0

Matrix Equations to solve a 3x3 system of equations

Example:
Write the matrix equation to represent the system, then use an inverse matrix to solve it. (Use a calculator)
5x - 2y + 4x = 0
2x - 3y + 5z = 8
3x + 4y - 3z = -11

Solving a 3 × 3 System of Equations Using the Inverse

Example:
Solve the system of equations using an inverse matrix. (Don’t use a calculator)
x + 2y + 2z = 5
3x - 2y + z = 6
2x + y - z = -1

Solving systems of Equations using Matrices
Using Inverse Matrices to evaluate a system of equations. (Use a calculator)

Example:
3x - 2y + z = 24
2x + 2y + 2z = 12
x + 5y - 2z = -31

Solving 3-Variable Systems - Matrix Method
Solving a system of equations with 3 variables.

Example:
4x + 2y - 2z = 10
2x + 8y + 4z = 32
30x + 12y - 4z = 24

This is a calculator that can help you find the inverse of a 3×3 matrix.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.