### Simultaneous Equations - interactive practice

In this tutorial, you will learn how to solve simultaneous equations by:

- Elimination Method
- Graphical Method
- Substitution Method
- Matrix Method

Equations that must be solved at the same time are simultaneous equations. They have two *unknowns.*

**E.g.**

2x + 3y = -9

x + 4y = 6

We use three different methods to solve simultaneous equations. They are:

- Elimination method
- Substitution method
- Graphical method
- Matrix method

#### Elimination Method

In this method, we must get rid of one variable in order to find the other.

**E.g.1**

x + y = 6 1

x - y = 2 2

If we add the two equations, we can remove y.

1 + 2 => 2x = 8

x = 4

Sub in 1 => 4 + y = 6

y = 2

Solutions are x = 4 and y = 2.

**E.g.2**

2x + y = 6 1

3x - 2y = 2 1

To remove y, multiply the first equation by 2 and then add the two equations together.

1 X 2 => 4x + 2y = 123

2 + 1 => 7x = 14

x = 2

Sub in 1

4 + y = 6

y = 2

The solutions are x = 2 and y = 2.

**E.g.3**

2x + 3y = 1 1

3x - 2y = 8 2

In this case, to eliminate y, the first equation must be multiplied by 2 and the second equation must be
multiplied by 3.

1 X 2 => 4x + 6y = 2 3

2 X 3 => 9x - 6y = 24 4

3 + 4 => 13x = 26

x =2

Sub in 1=> 4 + 3y = 1

-4 => 3y = -3

y = -1

The solutions are x = 2 and y = -1.

#### Substitution Method

We get y in terms of x or vice versa from one equation, and put that in the other.

**E.g.1**

x + y = 6 1

x - y = 2 2

From 1 => x = (6 - y)

Sub this in 2 => 6 - y - y = 2

6 - 2y = 2

-6 => -2y = -4

:--2 => y = 2

Sub in 1 => x + 2 = 6

x = 4

Solutions are x = 4 and y = 2.

**E.g.2**

2x + y = 6 1

3x - 2y = 2 2

From 1 => y = (6 - 2x)

Sub this in 2 => 3x - 2(6 - 2x) = 2

3x - 12 + 4x = 2

7x - 12 = 2

+ 12 => 7x = 14

x = 2

Sub in 1 => 4 + y = 6

-4 => y = 2

The solutions are x = 2 and y = 2.

**E.g.3**

2x + 3y = 1 1

3x - 2y = 8 2

From 1 - 3y => 2x = (1 - 3y)

x = (1 - 3y)/2

Sub in 2 => 3(1 - 3y)/2 - 2y = 8

X 2 => 3(1 - 3y) - 4y = 16

3 - 9y - 4y = 16

3 - 13y = 16

-3 => -13y = 13

y = -1

Sub in 1 => 2x - 3 = 1

+ 3 => 2x = 4

x = 2

The solutions are x = 2 and y = -1.

#### Graphical Method

In this method, two straight lines are drawn for each equation. Then the point where the two lines intersect at is noted. The coordinates of this point are the solutions of the equations.

**E.g.1**

2x + y = 8 1

y -x = 1 2

1 => y = 8 - 2x

2 => y = x + 1

x | y = 8 - 2x |

0 | 8 |

-1 | 10 |

2 | 4 |

x | y = x + 1 |

0 | 1 |

-1 | 0 |

2 | 3 |

Rearrange the two equations in the form of y = mx + c and draw two lines for them on the same grid.

The coordinates of the point of intersection are x = 3 and y = 2.

So, the solutions are x = 3 and y = 2.

#### Matrix Method

*This is ideal for those who do Further Mathematics(FP1) at A-Level. Matrices is part of Further Mathematics Syllabus. It is useful for advanced mathematics too.*

In this method, the **inverse matrix** of the matrix **P** must be found first. Then, matrix, **Q**, which gives the values of x and y could be found by **matrix multiplication.** The following image clearly shows how it is done.

The method can be extended to solve any pair simultaneous equations; all you have to do is rearranging them in matrix form.

#### Use of Matrix Method in Microsoft Excel

In the following animation, the method of solving simultaneous equation using Microsoft Excel is shown.

The coefficient of the two variables form a **matrix**, at first. Excel creates its **inverse matrix**. Then the matrix formed by the values of the right hand side are multiplied by the inverse matrix to find teh solutions.

Please follow the steps below to use MS Excel as a practising exercise:

- Generate simultaneous equations from the question generator at the bottom of the page.
- Use
**MINVERSE()**of MS Excel to generate the**Inverse Matrix.** - Use
**MMULT()**of MS Excel to multiply the inverse matrix and the value matrix. - The product of the two matrices gives the solutions of the simultaneous equations.

**Now, in order to complement what you have just learnt, work out the following questions:**

**Simultaneous Equation Generator**

Two random equations will appear below, when clicked