"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.
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.
3f + 2c = 280 (1)
f + 4c = 260 (2)
These two equations are both true at the same time, hence the
There are two methods of solving simultaneous
equations. Use the method which you prefer:
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
\ f = 60
price of fish is 60p
Substitute this value into (1):
3(60) + 2c =
\ 2c = 100
Therefore the price of chips is 50p
The method of substitution involves transforming one equation
into x = something or y = something and then substituting this into the other
Rearrange one of the original equations to isolate a
Rearranging (2): f = 260 - 4c
Substitute this into the other
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 x², y² or xy, we need to
use the method of substitution.
2xy + y = 10 (1)
x + y = 4 (2)
Take the simpler
equation and get y = .... or x = ....
from (2), y = 4 - x (3)
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 - 2x² + 4 - x - 10 =
\ 7x - 2x² - 6 = 0
\ 2x² - 7x + 6 = 0 (taking everything to the
other side of the equals sign)
- 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 =
Simultaneous equation can also be solved by graphical methods.
Copyright © Matthew Pinkney 2003