The problem of identifying linear static relations from noisy data is investigated under the assumptions of the Frisch scheme. The work presents an attempt to simplify the search for solutions of the related rank reducibility problem. The proposed approach is based on a relaxation of the original problem by means of algebraic manipulations of the non-linear equations which characterize the solution set. It is shown that this method can be advantageously used when particular conditions regarding the number of variables and the number of linear relations are satisfied.