The identity is a mathematical expression that is valid for all values of variables, while the equation is a mathematical expression that is valid only for certain values of variables.

Consider as example the following identity, such that:

`x^2 - y^2 = (x - y)(x + y)`

You should notice that the identity above holds for all real values of variables x and y.

Consider as example the following equation:

`xy - (x + y) + 1 = 0`

You need to notice that the identity `xy - (x + y) + 1 = 0` becomes valid only for `x = y = 1` , other real values of x and y being rejected.

Hence, you may say that the expression `xy - (x + y) + 1 = 0` represents an equation whose solutions are `x = y = 1.`

**Hence, equation is a mathematical sentence that is used to evaluate the values of a variable, while the identity, is a mathematical sentence that is used to emphasize the equivalence of two mathmatical expressions.**

An arithmetic expression that equates one set of conditions to another; for example, **A = B + C.**

identity :it is defined as an equality that holds regardless of the values of its variables