How to do Identities and Not Get Muddled like Spaghetti

Just when you thought you were finally getting GCSE Maths and starting to enjoy some sleep, safe in the knowledge that you're going to pass your exam...this symbol pops up:

That, my friend, is an Identity symbol and it usually comes up like this:

\[x\ +\ 1\ ≡\ 1\ +x\]

Or this:

Work out the values of a and b in the following identity:

\[5\left(7x+8\right)+3\left(2x+b\right)\ ≡\ \ ax\ +13\]

You are allowed a mild panic and a biscuit at this point.

The good news? They are easier than they look!

So what is an Identity?

It is a type of equation where both sides are always equal to each other.

For example an equation is only true for specific values of x .

3x + 5 = 8, is only true when x is 1.

However an identity is always true, regardless of the value of x, like in this one:

\[x\ +\ 1\ ≡\ \ 1\ +\ x\]

Go ahead and plug different numbers in there and you will see that both sides are always the same.

In the GCSE Maths Exam you will usually see an identity paired up with an algebraic equation (sometimes a quadratic) like the second example which is from the Nov 2018 paper.

Let's look at how to do a question like that:

\[5\left(7x+8\right)+3\left(2x+b\right)\ ≡\ \ ax\ +13\]

First we need both sides to look the same, something x + a number.

We can do that by multiplying out the brackets on the left hand side:

\[35x\ +\ 40\ +\ 6x\ +3b\ ≡\ \ ax\ +13\]

\[41x\ +\ 40\ +\ 3b\ ≡\ \ ax\ +13\]

Now we can compare.

The letter a in this question is simply the number of xs, or in math speak, the coefficient of x. So a is 41 - easy.

To find b, we need to do a little more work.

On the right hand side we have a constant, which is positive 13.

Because this is an identity, this means that the constant on the left hand side has to be equal to 13 too.

The constant on the left hand side looks a bit different at the moment: 40 + 3b.

This means:

\[40\ +\ 3b\ =13\]

So now we can solve that mini equation to find b:

\[3b\ =\ 13\ -40\ \]

\[3b=-27\]

\[b=-9\]

So a is 41 and b is -9.

Hope that helps!

Remember these questions come up fairly early in the exam so they are not usually too difficult. If you do get stuck on it in the exam, move on and come back to it when you've completed some other questions and your stress levels come down a bit.

Please do comment below if this was useful or if you have any more questions about Identities or GCSE Maths in general.