  In this paper, we consider the solution of ill-conditioned systems of linear
algebraic equations that can be determined imprecisely. To improve the
stability of the solution process, we "immerse" the original imprecise linear
system in an interval system of linear algebraic equations of the same
structure and then consider its tolerable solution set. As the result, the
"intervalized" matrix of the system acquires close and better conditioned
matrices for which the solution of the corresponding equation system is more
stable. As a pseudo-solution of the original linear equation system, we take a
point from the tolerable solution set of the intervalized linear system or a
point that provides the largest tolerable compatibility (consistency). We
propose several computational recipes to find such pseudo-solutions.
