Abstract: Using the combinatorics of Lakshmibai-Seshadri paths, we give explicit cancellation-free formulas for the action of multiplication by a dominant or antidominant line bundle, on the basis of structure sheaves of Schubert varieties, in the torus-equivariant K-theory of a Kac-Moody flag manifold. They are extensions of formulas of Pittie and Ram and Griffeth and Ram, who worked in finite-dimensional flag manifolds (and whose proofs had significant gaps for nonregular weights). We will mention evidence that this rule essentially determines the K-theory ring structure; it is known to do so in the finite-dimensional case. This is joint work with Cristian Lenart, who would probably be mad if I didn't mention that we also have versions of the Chevalley rule in terms of the Lenart-Postnikov alcove/lambda-chain model.