  Systems of fuzzy relation equations and inequalities in which an unknown
fuzzy relation is on the one side of the equation or inequality are linear
systems. They are the most studied ones, and a vast literature on linear
systems focuses on finding solutions and solvability criteria for such systems.
The situation is quite different with the so-called weakly linear systems, in
which an unknown fuzzy relation is on both sides of the equation or inequality.
Precisely, the scholars have only given the characterization of the set of
exact solutions to such systems. This paper describes the set of fuzzy
relations that solve weakly linear systems to a certain degree and provides
ways to compute them. We pay special attention to developing the algorithms for
computing fuzzy preorders and fuzzy equivalences that are solutions to some
extent to weakly linear systems. We establish additional properties for the set
of such approximate solutions over some particular types of complete residuated
lattices. We demonstrate the advantage of this approach via many examples that
arise from the problem of aggregation of fuzzy networks.
