# Simultaneous equations

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

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 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 = …’:

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:

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 two unknowns.  Since we’re going to be working with simultaneous equations, we should assign some letters to represent the unknowns:                                           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 add up to 75 years.  This means we can write another equation:                                                           This is a second piece of information, which is independent of the first.  So we have two unknowns, and two independent pieces of information.  This means we should be able to find unique values for the two unknowns.  We’ve got two equations:                                                                                                                       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 .  So in the second equation, instead of writing ‘j’, we can write ‘4m’:                                                           But , so:                                                         So Matthew is 15 years old.  Now, we just need to work out how old John is.  We know that , so:                                                           So John is 60 years old.