Using Matrices To Solve A System Of Equations Or Simultaneous Equations

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Types Of Matrices
Solving Systems of Equations or Simultaneous Equations using algebra
More Lessons On Matrices
Algebra Lessons

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

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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.

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How To Solve Simultaneous Equations Using Matrices?

Step by step solution.

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

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Simultaneous Equations - Matrix Method

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