The paper studies language equations of the type X lozenge L = R and L
lozenge Y = R, where L and R are given languages and lozenge is an in
vertible binary word (language) operation. For most of the considered
insertion and deletion operations, the existence of both a solution an
d a singleton solution to these equations proves to be decidable for g
iven regular L and R. In case L is a context-free language and R is a
regular one, the existence of a solution is generally undecidable. The
results can be extended to more complex linear equations, systems of
linear equations as well as for equations of higher degree.