AbstractThis work is part of a project on weight bases for the irreducible representationsof semisimple Lie algebras with respect to which the representation matrices ofthe Chevalley generators are given by explicit formulas. In the case of sln, thecelebrated Gelfand-Tsetlin basis is the only such basis known. Using the setupof supporting graphs developed by Donnelly, we present a new interpretation anda simple combinatorial proof of the Gelfand-Tsetlin formulas based on a rationalfunction identity (all the known proofs use more sophisticated algebraic tools). Aconstructive approach to the Gelfand-Tsetlin formulas is then given, based on asimple algorithm for solving certain equations on the lattice of semistandard Youngtableaux. This algorithm also implies certain extremal properties of the Gelfand-Tsetlin basis...