The Identity Hunt

A system of equations where the question asks for an unusual combination (x − y, x + y, x², etc.) rather than x or y individually. Or, an identity-flavored question about GCFs, sums, products.

Solving for x and y separately. Takes 4× longer and invites arithmetic errors.

1. Look at what's being asked. Is it x − y? x + y? Something asymmetric like 2x + 3y?
2. Look at the given equations. Can you ADD them, SUBTRACT them, or MULTIPLY one by a constant to directly produce what's asked?
3. Common identities to know cold:
4. • (a + b)² = a² + 2ab + b²
5. • (a − b)² = a² − 2ab + b²
6. • a² − b² = (a + b)(a − b)
7. • (a + b)(a − b) = a² − b²
8. If a question gives you (a + b) and (a − b) and asks for a² − b², you're 5 seconds away.

GRE rewards elegance. Every test has 2-3 questions that are 10 seconds if you spot the identity, 90 seconds if you don't.