The results of the study of improper linear programming problems are presented, in which the duality theory is essentially used and the approaches of I.I. Eremin (correction of incompatible constraints) and A.N. Tikhonov (creation of compatible systems of constraints equivalent in accuracy to given incompatible constraints). The problem of a stable solution to an approximate (and, possibly, improper) pair of mutually dual linear programming problems with a coefficient matrix of size \(m\times n\) is reduced to a Mathematical Programming problem of dimension \(m + n + 2\). The necessary and sufficient conditions for the existence of a solution and constructive formulas for its calculation are obtained. Computational experiments were carried out on a model Linear Programming problem with an approximate matrix and vectors of the right-hand side and the objective function, demonstrating the convergence of the obtained solutions to the normal solutions to direct and dual problems with a decrease in the level of data error.