# Equations and identities

An equation states that two expressions are equal in value, eg .

Solving an equation means finding the value or values for which the two expressions are equal. This means equations are not always true. In the example above, , the only correct solution for is 2.

An identity is an equation which is always true, no matter what values are substituted. is an identity because will always simplify to regardless of the value of . Identities can be written with the sign ≡, so the example could be written as .

### Example

Show that is the solution of the equation

BIDMAS means the multiplication is carried out before the addition:

Question

Are the following an identity or an equation?

• is an equation because the expression on the left of the equals sign cannot be rearranged to give the equation on the right. The solution to the equation is .
• is an identity because when you expand the bracket on the right of the equals sign, it gives the same expression as on the left of the equals sign.
• is an equation because the expression on the left of the equals sign cannot be rearranged to give the equation on the right. There is no solution for this equation – no matter what value of is substituted into the equation, the expression on the left will never have the same value as the expression on the right.