**What is an identity?**

**Identities** are a special type of equation.

They are **always true**, no matter what values are substituted in.

As this is such a special case, in identities we replace an equal sign with an identity sign which looks: **≡**

Let's see this in action now.

**e.g. (a + b)² = a² + 2ab + b²**

This is an identity, because whatever values we substitute for a and b, we always get a statement that is** true**.

Let's test this with a = 2 and b = 3...

On the left-hand side (LHS): (2 + 3)² = 5² = 25

On the right-hand side (RHS): 2² + 2×2×3 + 3² = 4 + 12 + 9 = 25

25 = 25

So (a + b)² = a² + 2ab + b² is definitely an** identity**.

We could also expand and simplify.

This would give us the same expression on the left as on the right of the identity sign.

**How to use an identity?**

At GCSE level, you might be asked to use the properties of **identities** to find **missing coefficients**.

**e.g. 5ax + 8 + 3(x - d) ≡ 18x + 14**

**Find the values of a and d.**

The highlighted sign is an identity sign, so this means that the LHS and the RHS will simplify to exactly the same expressions.

Let's expand and simplify the LHS first:

5ax + 8 + 3(x - d) = 5ax + 8 + 3x - 3d

Now let's remove x from the RHS, by factorising the left-hand terms with x:

5ax + 3x + 8 - 3d = x(5a + 3) + 8 - 3d

Therefore, 5a + 3 = 18 and 8 - 3d = 14.

So a = 3, d = -2

In this activity, we will learn to recognise **identities** and use their unique properties to fine the value of terms in expressions.