Simultaneous equations are fun.

No, really, they are, especially the simple ones.

e.g.
5x + y = 17
6x + 2y = 22

They are called simultaneous, I suppose because you get 2 equations to muck about with, or maybe because there are two variables, who knows? No one, that's who.

You are not supposed to solve both equations at exactly the same time. Oh, no, that would be far too hard, even for the boffs at the front of the class, what you do is look at them both and decide which is easiest to do and then start with that one.

Here's a bit of maths logic, let's abbreviate that to 'magic' - hah! See what I did there, well, like all magic it's really trickery. (Except Harry Potter of course that's all true.)

The trick is to turn an equation that has two variables into an equation that has only one, then they are really easy eg x=3, see! You worked x out straight away!


Let's get down to it then. Imagine a REALLY easy simultaneous equation (shall we call it a simation... err, no), this is so easy it's no fun:

5x + y = 13
y = 3

we know y = 3 that's obvious straight away (even to those boys at the back)

Now we can put the 3 in where the y is in the first equation (as they are the same) and get

5x + 3 = 13

although we could work out x in our heads - without having to be a mega boffin. we are going to follow a process that will help us when these equations get more difficult.

what we do is look for things that are the same on both sides.

there are 5 'x' s on the left but none on the right so that's no good, but there are simple numbers on both sides, 3 is on one side and 13 is on the other, so what we do now is take away equal amounts from each side to keep it our equation balanced. In this case we can only take 3 away as that's all there is on the left, and so we take 3 from the right too.

so now we have

5x = 10

Don't jump ahead now! We all now how clever you are.

Once again we look for things that are the same on both sides, this time we move to the next level, not just things that we can take away, but things we can divide by work as well. Before long we realise we can divide both sides by 5. Now we have

x = 2

And that is the answer - or should I say answers:

y = 3 and x = 2.

You have learnt the basic principles (I think you deserve a gold star in your sticker book) and now I'm going to show you how to apply them to the 'fun' equation at the top of the page ie

5x + y = 17
6x + 2y = 22

Looking at these two the easiest one to work with is the first as there is no multiplier on the y. What we are hoping to do is to make an equation similar to y = 3 in the solution we just worked out, but in this case y will be equal to something else, it doesn't matter what really, as long as we keep the equation balanced and only y is on one side.

We have kept equations balance by subtraction and division, these are the two ways we will use today, but now I'm adding a new trick. This time we're going to subtract 5x from both sides, 'Oh are you?' I hear you say, 'Well there aren't 5 'x's on the right' - 'oh yes I am!' I reply, because I'm going to use negative numbers!

Just watch....

5x + y - 5x = 17 - 5x

totting up the 'x's on the left we get nothing, zilch, nowt, zero, so:

y = 17 - 5x

Ta daaaa! We have our equation with just y on one side.

Now let's replace the y in the second equation with the equivalent we have just worked out (17 - 5x).

6x + 2(17 - 5x) = 22

let's expand those brackets (hey, get with the maths terminology!)

6x + 34 - 10x = 22

add up the 'x's on the left (-10 + 6 = -4, keep up!)

34 - 4x = 22

subtract 22 from both sides (am I going too fast for you?)

12 - 4x = 0

now add 4x to both sides

12 = 4x

and finally divide both sides by four

3 = x

or

x = 3

now we know what x is we can put it in the formula with just y on one side, i.e.

y = 17 - 5x

y = 17 - (5 x 3)

y = 17 - 15

so

y = 2

Phew, wasn't that fun!

Well they were pretty easy but the principles hold for all simultaneous equations... and here's a top tip.

Exam questions involving simultaneous equations very rarely use high numbers or non integers (You at the back! yes you! integers means whole numbers). If either of these things appear in your calculations check your workings - the question will be designed to test your logic not your calculator!

Now you are a boff, level 1.