identity equation

Most equations are picky: they hold for only certain values of the variable. An identity equation is the opposite — it is true no matter what number you plug in. It is less a puzzle to solve than a fact that is always so.

When you try to solve an identity, the variable cancels out entirely and you are left with a statement that is plainly true, like 0 = 0 or 5 = 5. That is the telltale sign: the algebra reduces to something true with no variable remaining.

Its solution set is every real number, so you say it has infinitely many solutions. This is the natural cousin of a contradiction, which collapses instead to a false statement and therefore has no solution. Recognizing both keeps you from chasing an answer that is either everywhere or nowhere.