Trevor Pythagoras

Simultaneous equations are two or more equations that are all related because they contain the same variables. To solve these equations you must use information from both equations to find a set of values for the variables that hold true for both equations.

You can only solve simultaneous equations if there at least as many different equations (being different meaning that the two equations cant be re-arranged or simplified into each other) as there are variables in the equations. In order to solve the equations we want to eliminate all the variables but one so that we are left with a single equation that can be solved normally.

There are several ways of solving these equations. The method of solving using substitution is shown below using an example.

Solve using substitution – with an example

Lets consider the equations
(1)- y + 3x = 12 + 4y
(2)- 3x = 7y – 6 + 4x

First of all we must simplify the equations by gathering all the similar terms together and cancelling through any common factors. So we get
(1)=> 3x = 12 + 3y => x = 4 + y
(2)=> 6 = 7y + x

We would now normally arrange one of these equations so that either x or y is the subject of the equation however in this example equation 1 naturally has x as the subject.

We can now substitute the expression for x in equation 1 (4+y) into equation 2 to get

6 = 7y + 4 +y

Again we must now simplify this to get

2 = 8y

and then we can solve it as a single equation to find
y=1/4

We can now substitute this value for y into either of the above equations to get a value of x. Substituting into 1 gives

x = 4 + 1/4
x=5/4

So that we now have our values of x and y that satisfy both of the equations

x=5/4 and y=1/4

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