  We deal with solutions of classical linear equations ax=b and ya=b, applying
a particular lattice valued fuzzy technique. Our framework is a structure with
a binary operation (a groupoid), equipped with a fuzzy equality. We call it a
fuzzy quasigroup if the above equations have unique solutons with respect to
the fuzzy equality. We prove that a fuzzy quasigroup can equivalently be
characterized as a structure whose quotients of cut-substructures with respect
to cuts of the fuzzy equality are classical quasigroups. Analyzing two
approaches to quasigroups in a fuzzy framework, we prove their equivalence. In
addition, we prove that a fuzzy loop (quasigroup with a unit element) which is
a fuzzy semigroup is a fuzzy group and vice versa. Finally, using properties of
these fuzzy quasigroups, we give answers to existence of solutions of the
mentioned linear equations with respect to a fuzzy equality, and we describe
solving procedures.
