# Simultaneous equations

Say I’ve got a really, really simple equation like this:

_{}

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Can I work out what the value of x is? Yes, of course. x is equal to 4. What about this equation:

_{}

Can I work out what x is equal to? Yes. But first I have to work out what 4 + 2 is:

_{}

x is equal to 6. In both of these equations, there
was __one__*unknown value* – the ‘x’. By working with the
equation, I was able to find out what the value of ‘x’ was. In the first case,
I didn’t have to do much – I just had to read the equation. In the second
case, I had to do a simple addition before I could work out what ‘x’ was.

Now how about this equation:

_{}

Now I have __two__*unknown* values
– the ‘x’ *and* the ‘y’. Can I work out what the values of ‘x’ and ‘y’
are?

Well, I could rearrange this equation so it was something like ‘x = …’:

_{}

But this isn’t much use – I still don’t know what value x is, all I know is that it’s “10 minus y”. Perhaps I can rearrange the original equation so I get ‘y = …’:

_{}

Arrrggg! I can’t work out what the value of y is
either – all I know is that it’s “10 minus x”. But hold on a second – I do
have an *expression* for what x is. The expression is the one I got
before; _{}.
How about if I substitute this into the latest equation like this:

Here’s my ‘y = …’ equation:

_{}

Instead of the ‘x’, let me write ‘10 – y’, because we know this is what ‘x’ is equal to:

This means we get something like this:

_{}

Wow! We’ve just proved that ‘y = y’. In other
words, doing this hasn’t given us any extra information on how to work out what
‘x’ and ‘y’ are. And this is quite simply because *we don’t have enough
information to work out what they are*.

We have *two* unknown values – ‘x’ and ‘y’.
However we only have *one* piece of information about them – we have *one*
equation saying ‘x + y = 10’. Now, we did rearrange this ‘x + y’ equation into
some different layouts, but it’s still the *same piece of information*, so
these rearrangements don’t count as extra pieces of information.

To work out the unique values of the unknowns in
equations, you need at least as many *different* pieces of information as
there are unknown values. In our case we don’t have enough information, so all
we can do is work out possible values that x and y could have:

· If x = 5, then y = 5

· If x = 2, then y = 8

· If x = 9, then y = 1

· If x = –5, then y = 15

We could go on forever. We could even use the _{} equation format
to draw a straight line graph showing all the values of x and y we could have:

Any point along the line is a valid set of (x, y)
points that make the equation true. For instance, the point where the line
crosses the x-axis is at x = 10 and y = 0, which are valid values for _{}.

So say we got given an *extra piece of
information*, something like this:

_{}

Now we have *two *pieces of information, and
we have *two *unknown values. The two equations we have now are normally
called *simultaneous equations*. We should be able to work out what they
are now.

Remember how earlier we had this equation _{}, and we tried to
replace the ‘x’ with something else. Well, let’s try doing that again, but
this time let’s use our new piece of information, _{}:

This means that we get this:

_{}

But because x = 2y:

_{}

Bingo! We’ve finally got a value for y. Now it’s easy to work out what x is. We have an equation in the form ‘x = …’:

_{}

Handy Hint #1 - Picking the best equation to finish off a question with

When you’re in an exam, you’re often pushed for time. It’s always good to be able to do a question as quickly as possible. When you’re working out the value of the last unknown in a simultaneous equations question, you can save time by picking the best equation to use.

Halfway through the last question, I’d
already worked out that _{}. Now I needed to solve what ‘x’
was. I had the following equations I could use:

_{}

_{}

Which one is the best to use? Well, since I’m trying to find out what ‘x’ is, I need an equation which is in the form, “x = …” The first equation isn’t in this form – I’d have to move some bits around and change some signs before it would be. But the second equation is just right – it’s already in the form, “x equals …” All I have to do is plug the value of ‘y’ straight into the right hand side and I’ll have my answer.

So make sure you always look at all your options, and pick the one which is the easiest to use. You don’t want to make your job any harder than it has to be!

### Independent pieces of information

Say I get told that y is 4 times larger than x. In mathematical form, I would write this like this:

_{}

Now, say I get told another piece of information, that 2 lots of y is 8 times larger than x. In mathematical form I would write this like:

_{}

Now, is this second piece of information really any different to the first piece of information? It looks different now, but what about if I simplify it by dividing both sides by 2:

_{}

Aha! When we simplify it, it ends up looking just
like the first piece of information. These two pieces of information *are
not independent*. For two pieces of information to be independent, there
has to be no way you can manipulate or simplify one to get the other. For
instance, these two equations are independent because there is no way I can
change one so that it looks like the other:

_{}

Handy Hint #2 - Working out whether you can find unique answers

One of the things you need to be able
to do is work out whether, given the information in a question, you can work
out *unique* values for the unknowns. For instance, say you’re given:

_{}

You should be able to take one look at this and say straight away that you can’t find unique values for ‘x’ and ‘y’. A good method to use is to look at the information given and follow these three steps:

1. How many unknowns are there in the question?

2. How many independent pieces of information do we have?

3. If
your answer to part 2 is the same size or larger than your answer to part 1,
you should be able to find *unique* values for your unknowns.

Say we have _{}. We can follow this 3 step
procedure to see whether we can solve for the unknowns.

There are 2 unknown values – ‘x’ and ‘y’. So we can write:

There are *two*
unknowns – ‘x’ and ‘y’.

We only have *one* piece of information however
– the equation _{}.

There is *one* piece
of information in the question.

For step 3, we compare the number of pieces of
information we have with the number of unknowns. Because we’ve got more
unknowns than pieces of information, we can not find *unique* values for
the unknowns.

Of course, these questions can be more complicated
than this. One of the trickiest parts is to *translating* the words in a
question into a more mathematical form. Here’s a sample word question:

John is 4 times as old as his grandson Matthew. Together their ages add up to 75 years. What are their ages? |

Solution |

First up, we need to work out how many unknowns
there are in the question. This part’s pretty easy – the unknowns are the
ages of the two people. So we have Let ‘j’ = the age of John in years. Let ‘m’ = the age of Matthew in years. Second thing to do is work out how many independent pieces of information we have in the question. The first sentence gives us one piece of information – John is 4 times as old as his grandson Matthew. So we can write something like: This is one piece of information. If we keep
reading the question, we come across another piece of information – together
their ages This is a second piece of information, which is In the second equation, we’ve got a ‘j’ and an
‘m’. By itself, this equation won’t give us unique values for ‘j’ and ‘m’.
But thanks to the first equation, we know that But So Matthew is 15 years old. Now, we just need to
work out how old John is. We know that So John is 60 years old. |