equivalent equations

Two equations are equivalent the way two roads to the same town are equivalent: they may look different, but they end at exactly the same place. For equations, that shared destination is their solution set.

Equivalent equations have precisely the same solutions — every value that solves one solves the other, and nothing more. So x + 2 = 5 and x = 3 are equivalent: both are satisfied by 3 alone.

Solving works by stepping from one equation to a simpler equivalent one, again and again, until the variable stands alone. The legal moves — adding the same quantity to both sides, or multiplying both sides by a nonzero number — are exactly the ones that preserve equivalence. Multiplying by zero is forbidden because it can manufacture false solutions.