# Using Matrices to Solve a System of Equations or Simultaneous Equations

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

Related Topics:

Solving Systems of Equations or Simultaneous Equations using algebra

More Algebra Lessons

## Matrices and Inverse Matrices

Simultaneous equations or system of equations of the form

*ax + by = h *and

*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}, we get

A^{-1}A Y = A^{-1}B

I Y = A ^{-1}B ( AA ^{-1}= I, where I is the identity matrix)

Y = A ^{-1}B ( IY = Y, any matrix multiply 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:

2*x* – 2*y* – 3 = 0

8 *y* = 7*x* + 2

* Solution: *

**Step 1:** Write the equations in the form *ax + by = c *

2*x* – 2*y* – 3 = 0 ⇒ 2*x* – 2*y* = 3

8*y* = 7*x* + 2 ⇒ 7*x* – 8*y* = –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

## Videos

The following video shows how matrices can be used to solve a system of equations.

Using the inverse of a matrix to solve a system of equations.

Using a Matrix Equation to Solve a System of Equations

This video shows how to solve a system of equations by using a matrix equation. The graphing calculator is integrated into the lesson.

Solving Simultaneous Equations Using Matrices.

Simultaneous Equations - Matrix Method.