"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 x², y² 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  2x² + 4  x  10 =
0 \ 7x  2x²  6 = 0 \ 2x²  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
