  Systems of integration-by-parts identities play an important role in
simplifying the higher-loop Feynman integrals that arise in quantum field
theory. Solving these systems is equivalent to reducing integrals containing
numerator products of irreducible invariants to a small set of master
integrals. I present a new approach to solving these systems that finds direct
reduction equations for numerator terms of a given Feynman integral. As a
particular example of its power, I show how to obtain reduction equations for
arbitrary powers of irreducible invariants, along with their solutions.
