Back-Solving: Plugging In the Answer Choices

Solving an equation algebraically and testing numbers until one fits are both valid paths to the same answer. Back-solving wins on speed when the algebra is slow or error-prone, like a quadratic that doesn't factor nicely, because plugging in a number is pure arithmetic with no setup. It also wins when you're unsure of your algebra, since a number that checks out is proof, not a guess.

The trick to doing it fast: start with one of the two middle values among the four ordered choices, not the first one listed. Answer choices on numeric problems are almost always arranged in order, so testing a middle value tells you not just whether it's right, but whether the true answer is bigger or smaller. If your test value gives a result that's too small, you've also eliminated every choice smaller than it in one move, cutting what's left to check roughly in half. One more test from the remaining side usually finds the answer in two tries instead of four.

This only works when the choices themselves are candidate numbers you can drop into the problem. If the choices are expressions in x, there's no single value to test against the stem, since each choice is a different rule for turning any x into an output, not a specific number that might satisfy the equation. In that situation the faster move is picking a convenient number for x, computing the real result by hand, and seeing which expression reproduces it. Back-solving and picking-numbers are cousins: one tests the answer, the other tests the setup.