GRE-Style: System of Equations with Constraints

GRE Quantitative · Axiom Academy

EXAMPLE GRE-Style: System of Equations with Constraints Using an extra constraint to pick the right solution out of a system, then making a quantitative comparison. x and y are positive integers such that x + y = 12 and xy = 32 . Also, x > y . Nice work — you used a constraint to eliminate an extra root and land on a single valid comparison. Substitute to collapse the system: a sum-and-product system in two variables ( x+y=s , xy=p ) always reduces to one quadratic in a single variable. A quadratic can hand you two algebraically valid pairs: both (x,y)=(4,8) and (8,4) satisfy the original equations — the algebra alone doesn't pick one. The constraint is the real work of the problem: " x > y " (plus "positive integers") is not throwaway wording — it's what makes the comparison well-defined. GRE quantitative-comparison problems often hide the deciding condition in a short extra clause like this. Result: only (x,y) = (8,4) satisfies every condition, so x - y = 4 < 5 — Column B is greater. Whenever a system produces more than one algebraic solution, scan the problem for an extra constraint before you compare anything — it's usually there to narrow two possibilities down to one.

This is the written version of the interactive lesson above. See the full GRE Quantitative course.