  In this paper, we present methods for solving a system of linear equations, $
AX=b $, over tropical semirings. To this end, if possible, we first reduce the
order of the system through some row-column analysis, and obtain a new system
with fewer equations and variables. We then use the pseudo-inverse of the
system matrix to solve the system if solutions exist. Moreover, we propose a
new version of Cramer's rule to determine the maximal solution of the system.
Maple procedures for computing the pseudo-inverse are included as well.
