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:

  1. Elimination method
  2. Substitution method
  3. Graphical method
  4. 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

xy = 8 - 2x
08
-110
24
xy = x + 1
01
-10
23

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

two-graphs

 

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.

matrix method for simultaneous equations

 

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.

Solving simultaneous equations with MS Excel and matrices

 

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

  1. Generate simultaneous equations from the question generator at the bottom of the page.
  2. Use MINVERSE() of MS Excel to generate the Inverse Matrix.
  3. Use MMULT() of MS Excel to multiply the inverse matrix and the value matrix.
  4. 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





 

 

 

 

Resources at Fingertips

This is a vast collection of tutorials, covering the syllabuses of GCSE, iGCSE, A-level and even at undergraduate level. They are organized according to these specific levels.
The major categories are for core mathematics, statistics, mechanics and trigonometry. Under each category, the tutorials are grouped according to the academic level.
This is also an opportunity to pay tribute to the intellectual giants like Newton, Pythagoras and Leibniz, who came up with lots of concepts in maths that we take for granted today - by using them to serve mankind.

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