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.