  A linear vector equation in two unknown vectors is examined in the framework
of tropical algebra dealing with the theory and applications of semirings and
semifields with idempotent addition. We consider a two-sided equation where
each side is a tropical product of a given matrix by one of the unknown
vectors. We use a matrix sparsification technique to reduce the equation to a
set of vector inequalities that involve row-monomial matrices obtained from the
given matrices. An existence condition of solutions for the inequalities is
established, and a direct representation of the solutions is derived in a
compact vector form. To illustrate the proposed approach and to compare the
obtained result with that of an existing solution procedure, we apply our
solution technique to handle two-sided equations known in the literature.
Finally, a computational scheme based on the approach to derive all solutions
of the two-sided equation is discussed.
