Using Matrices To Solve A System Of Equations Or Simultaneous Equations

In these lessons, we will learn how to solve Systems of Equations or Simultaneous Equations using Matrices.

How To Solve Matrix Equations

The following diagrams show how to solve a systems of equations using matrices. Scroll down the page for more examples and solutions.

Simultaneous equations or system of equations of the form:
ax + by = h
cx + dy = k
can be solved using algebra.

Simultaneous equations can also be solved using matrices.

First, we would look at how the inverse of a matrix can be used to solve a matrix equation.

Given the matrix equation AY = B, find the matrix Y.

If we multiply each side of the equation by A-1 (inverse of matrix A), we get

A-1A Y = A-1B
I Y = A -1B    (AA -1 = I, where I is the identity matrix)
Y = A -1B       (IY = Y, any matrix multiplied with the identity matrix will be unchanged)

Matrices & Systems of Equations

Example:
Using matrices, calculate the values of x and y for the following simultaneous equations:
2x – 2y – 3 = 0
8 y = 7x + 2

Solution:
Step 1: Write the equations in the form ax + by = c
2x – 2y – 3 = 0 ⇒ 2x – 2y = 3
8y = 7x + 2 ⇒ 7x – 8y = –2

Step 2: Write the equations in matrix form.

Step 3: Find the inverse of the 2 × 2 matrix.
Determinant = (2 × –8) – (–2 × 7) = – 2

Step 4: Multiply both sides of the matrix equations with the inverse.

So, x = 14 and y = 12.5

How Matrices Can Be Used To Solve A System Of Equations?

Using the inverse of a matrix to solve a system of equations.
3x + 2y = 7
-6x + 6y = 6

How To Use A Matrix Equation To Solve A System Of Equations?

This video shows how to solve a system of equations by using a matrix equation.
AX = B
A-1AX = A-1B
IX = A-1B
X = A-1B

A 2 x 2 example and a 3 x 3 example are given.

Example:
Solve the system using a matrix equation
3x - y = 5
2x + y = 5

Example:
Solve the system using a matrix equation
x - 3y + 3z = -4
2x + 3y - z = 15
4x - 3y - z = 19

The graphing calculator is integrated into the lesson.

How To Solve Simultaneous Equations Using Matrices?

Step by step solution.

Example:
2x + 3y = 8
x - 2y = -3

Simultaneous Equations - Matrix Method

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.