# Concept of Identity

## Definition

Identity: An identity is an equality, which is true for all values of the variables in equality.

## Notes

### Identity:

Consider the equality (a + 1)(a + 2) = a2 + 3a + 2
We shall evaluate both sides of this equality for some value of a, say a = 10.
For a = 10,
LHS = (a + 1)(a + 2) = (10 + 1)(10 + 2) = 11 × 12 = 132
RHS = a2 + 3a + 2 = 102 + 3 × 10 + 2 = 100 + 30 + 2 = 132
Thus, the values of the two sides of the equality are equal for a = 10.

Let us now take a = – 5
LHS = (a + 1)(a + 2) = (– 5 + 1)(–5 + 2) = (– 4) × (–3) = 12
RHS = a2 + 3a + 2 = (–5)2 + 3 (–5) + 2 = 25 – 15 + 2 = 10 + 2 = 12
Thus, for a = – 5, also LHS = RHS.

We shall find that for any value of a, LHS = RHS. Such equality, true for every value of the variable in it, is called an identity.
Thus,
(x + y)2 = x2 + 2xy + y2
The above equation is true for all possible values of x and y, so it is called an identity.

An equation is true for only certain values of the variable in it. It is not true for all values of the variable. For example, consider the equation.

For example the equation,
3x = 12 is true only when x = 4, so it is an equation, but not an identity. In fact, when we see an equation like that, we are usually trying to solve it. That is, find the single value of x that makes the equation true.
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