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

Related Topics:
Solving Systems of Equations or Simultaneous Equations using algebra
More Algebra Lessons

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

### How to solve Matrix Equations?

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 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

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

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