Simultaneous Equations 1

Simultaneous Equations can be a bit of a mystery to some students. But that needn't be the case if you understand what they are about. The word "simultaneous" means "at the same time". With simultaneous equations, there are two lines (or a line and a curve) which cross each other and the crossing point is the "simultaneous" bit, i.e. at the crossing point, the x value for both lines is the same and the y value for both lines is the same.

What we are doing in simultaneous equations is finding out this  x value and y value, given the equations of two lines (or at a higher level, the equation of one line and a curve).

You might be asked to solve something like this:

                                     2x + y = 4

                                        x - y = 5

or this:                         3x + 2y = 14

                                      5x - 2y = 18


As in the Introduction to Equations, we will start with something easy. Let us say that  x = 1   and   y = 2.

Now we can make two simultaneous equations with these x and y values:

                                   x + y = 3                            (1)

and                           2x - y = 0                            (2)

Just check that you have followed that.

We will now solve these equations (even though we know the answers!)

We can add these two equations together, like this:

                    x + y + 2x - y = 3 + 0

Notice that the +y and -y cancel out and we have an equation just in x:

                   x + 2x = 3

i.e.                    3x = 3

and so               x = 1

The next step is to use either equation (1) or equation (2) to solve for y.

Using (1) we have   x + y = 3

As x = 1, this becomes    1 + y = 3

i.e.                                             y = 2

The final step is to use the other equation, i.e. (2), to check our answer.

                                      (2) says  2x - y = 0

Is this true?

                                      2 X 1 - 2 = 0

Yes, it is true and so we know that our answers are correct.


When we added the above two equations together, we did so in one long line. This was to make clear what we were doing. The more usual way of writing this is as follows:

                                          x + y = 3

                               +        2x - y = 0

                                         -------------

                                         3      = 3

and so                                x      = 1


We can now proceed as before to find the value of y, given that x = 1.

Equation (1)  says   x + y = 3

In place of the x we write 1:

                                   1 + y = 3

and so it is clear that y = 2.

Again, as before, we now substitute both values into equation (2) to check that our answers are correct.

                                  2x - y = 0

                                  2 X 1 - 2 = 0

Note: in equations, we use an italic x for the letter x and we use X to mean multiply.