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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.

### Matrices & Systems of Equations

**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^{-1}AX = A^{-1}B

IX = A^{-1}B

X = A^{-1}B

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**

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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.

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^{-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)

** 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

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

3x + 2y = 7

-6x + 6y = 6

This video shows how to solve a system of equations by using a matrix equation.

AX = B

A

IX = A

X = A

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.

Step by step solution.

Example:

2x + 3y = 8

x - 2y = -3

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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