equivalent expressions

Two expressions are equivalent if they are two recipes for the same result — like “double it then add six” versus “add three then double.” They may look different on the page, yet they produce identical output no matter what number you feed in.

Formally, two expressions are equivalent when they give the same value for every choice of the variables involved. So 2(x + 3) and 2x + 6 are equivalent: substitute any number for x and both yield the same answer. Equivalence is what lets you simplify, expand, or factor freely — every legal step replaces an expression by an equivalent one.

Equivalence demands agreement at every input, not just at one. 2x + 6 and 2x + 7 happen to be unequal everywhere, so that is easy. But x^2 and 2x agree at x = 0 and x = 2, yet differ elsewhere — agreeing at a couple of points is far from being equivalent. One mismatch anywhere is enough to disqualify them.