Simultaneous Equations

Introduction

Simultaneous equations are two or more equations which are true for two or more unknowns. For example, x + y = 4 and x - 2y = 1 are simultaneous equations which are true for x = 3 and y = 1 (these satisfy both equations). When there are two unknowns, as there are here (x and y), then two equations are needed to find the unknowns. When there are 3 unknowns, 3 equations are needed- in fact in order to solve simultaneous equations, in general there must be as many different equations as there are unknowns.

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

There are two methods of solving simultaneous equations. Use the method which you prefer.

Method 1: Elimination

In this method we add or subtract the equations to/from one another.

Doubling (1) gives:
6f + 4c = 560 (3)
(3)-(2) is 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

Method 2: Substitution

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
\ 2x² - 7x + 6 = 0
\ (2x - 3)(x - 2) = 0
\ either 2x - 3 = 0 or x - 2 = 0
\ 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

Using Graphs

You can solve simultaneous equations by drawing graphs of the two equations you wish to solve. The x and y values of where the graphs intersect are the solutions to the equations.

Example

Solve the simultaneous equations 3y = -2x + 6 and y = 2x -2 by graphical methods.
From the graph below, y = 1 and x = 1.5 (approx.). These are the answers to the simultaneous equations.