Technique Encyclopedia

Strategic Substitution

What it is

Strategic substitution means picking easy special values — often 0, 1, or -1 — to plug into a general statement, functional equation, or identity that's claimed (or asked) to hold for all valid inputs. A well-chosen substitution can extract the needed information in a single step, where full algebraic manipulation would take much longer.

Signals that suggest using it

A functional equation or identity is stated to hold for all values of one or more variables.
The problem asks for a specific value (like f(0), or a constant k) rather than a general formula.
An expression's answer is claimed to be independent of a variable — a special case can verify this quickly.

When it's effective

Fastest when a special value (0, 1, or matching two variables) makes most of an expression vanish or simplify dramatically, isolating exactly the quantity being asked for.

When it's not effective

A single substitution only confirms a value is consistent with one case — if a claim is supposed to hold identically for all inputs (not just verify a specific numeric answer), a lucky match at one value doesn't prove it in general; genuine algebraic verification is still needed for identities.

Simple example

Problem

A function satisfies f(x+y) = f(x) + f(y) for all real x, y, and f(1) = 3. Find f(0).

Solution

Substitute x = y = 0: f(0) = f(0) + f(0), which simplifies to f(0) = 0.

AMC-style example

Problem

If (x+1)² − (x−1)² = kx for all real x, find k.

Solution

Substitute x = 1: (2)² − (0)² = 4 = k(1), so k = 4. (Checking: expanding the left side directly gives 4x for every x, confirming k = 4 is not just a coincidence at x = 1.)

Testing Small Cases Working Backwards