Simultaneous equations are equations that share the same variables. Typically, we want to find both variables. Here's how.

## If both equations are linear

Imagine the two equations

3x - 7y = 1
y = 2x - 8

We want to find the value of x and the value of y.

3x - 7y = 1
y = 2x - 8

First, we rearrange the first equation to make y the subject. That goes like this -

3x - 7y = 1
3x - 1 = 7y
(3x - 1)/ 7 = y

So far so good. Now we have two equations for y. The next logical thing to do is put them together, like this -

(3x - 1)/ 7 = 2x - 8

Now, we solve this for x. We will then have a value for x. Like this -

3x - 1 = 7( 2x - 8 )
3x - 1 = 14x - 56
-1 = 11x - 56
11x = 55
55 / 11 = x
x = 5

x is 5! Now, to get y, we simply subsitute this value for x back into one of the original equations, like so -

y = 2x - 8
y = (2 * 5) - 8
y = 10 - 8
y = 2

Voila, we now have values for x and y! This method is the hardest of the two to get to grips with, but the simplest once you master it.

### If only one equation is linear

If only one of the equations in a simultaneous pair is linear, then the above method becomes a little different. Imagine one is linear and one is quadratic, like this -

y = 3x + 2
y = x2 + 2x + 2

As both of these are equations for y, we can put them together like this -

3x + 2 = y = x2 + 2x + 2

Then we can rearrange it into a quadratic equation, like this -

0 = x2 - x

The solution to this simple quadratic can then be found by completing the square, the quadratic formula or factorising. Then, when you have the two solutions to the quadratic equation, you simply plug these values for x back into one of the original equations to get values for y. In this case, when x = 1, y =5 and when x = 0, y = 2.

### Graphs

It is also possible to find the solutions to simultaneous equations by plotting graphs of them. The points where the two graphs intersect one another are the solutions.