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:

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

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

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