GCSE Revision Notes, Resources      

GCSE Maths > Algebra - Simultaneous Equations

"Simultaneous equations" are two or more equations which have two or more unknowns to be found. At GCSE, it is unlikely that you will have more than two equations with 2 values (x and y) which need to be found.

Example

A man buys 3 fish and 2 chips for 2.80
A woman buys 1 fish and 4 chips for 2.60
How much are the fish and how much are the chips?

First form the equations. Let fish be f and chips be c.
We know that:
3f + 2c = 280 (1)
f + 4c = 260 (2)

These two equations are both true at the same time, hence the name 'simultaneous'.
There are two methods of solving simultaneous equations. Use the method which you prefer:

Elimination

The method of elimination involves manipulating the two equations so that one can be added/ subtracted from the other to leave us with an equation with only one unknown.

In our above example:

Doubling (1) gives:
6f + 4c = 560 (3)
(3)-(2) gives 5f = 300
\ f = 60
Therefore the price of fish is 60p

Substitute this value into (1):
3(60) + 2c = 280
\ 2c = 100
c = 50
Therefore the price of chips is 50p

Substitution

The method of substitution involves transforming one equation into x = something or y = something and then substituting this into the other equation.

So,

Rearrange one of the original equations to isolate a variable.
Rearranging (2): f = 260 - 4c
Substitute this into the other equation:
3(260 - 4c) + 2c = 280
\ 780 - 12c + 2c = 280
\ 10c = 500
\ c = 50
Substitute this into one of the original equations to get f = 60 .

Harder simultaneous equations

To solve a pair of equations, one of which contains x, y or xy, we need to use the method of substitution.

Example

2xy + y = 10 (1)
x + y = 4 (2)
Take the simpler equation and get y = .... or x = ....
from (2), y = 4 - x (3)
this can be substituted in the first equation. Since y = 4 - x, where there is a y in the first equation, it can be replaced by 4 - x .
sub (3) in (1), 2x(4 - x) + (4 - x) = 10
\ 8x - 2x + 4 - x - 10 = 0
\ 7x - 2x - 6 = 0
\ 2x - 7x + 6 = 0 (taking everything to the other side of the equals sign)
\ (2x - 3)(x - 2) = 0
\ either 2x - 3 = 0 or x - 2 = 0
therefore x = 1.5 or 2 .

Substitute these x values into one of the original equations.
When x = 1.5, y = 2.5
when x = 2, y = 2

Simultaneous equation can also be solved by graphical methods.

Copyright Matthew Pinkney 2003

Site Search:


Copyright © 2004-2013 revisioncentre.co.uk In association with: Computer Science Revision | Revision Links and Tutors | EFL Teaching / Learning
Navigation: Home | Contact | KS2 Maths | Advice | Parents | Games | Shop | Links |
GCSE Subjects: Biology | Chemistry | Computing | English | French | Geography | German | History | Maths | Physics | RE | Spanish
Web Hosting by Acuras