contradiction

Some equations ask for the impossible. A contradiction is an equation that no number can satisfy — like demanding a value that is simultaneously two different things. Try as you might, no substitution will ever make it true.

You spot one the same way you spot an identity, by watching what happens as you solve. The variable cancels out, but this time you are left with a false statement, such as 0 = 5 or 3 = −1. The arithmetic itself refuses to be true.

Its solution set is empty: there are no solutions at all. This is the mirror image of an identity equation, which reduces to something always true and has infinitely many solutions. In systems of equations, a contradiction is the signature of an inconsistent system — two lines that never meet.