We consider a two-sided vector equation that is defined in terms of tropical algebra as Ax=By, where A and B are given matrices, x and y are unknown vectors. We propose a new procedure to solve this equation, which is based on the minimization of the distance between vectors of tropical vector spaces generated by the columns of the given matrices. The procedure produces a pair of vectors that provide the minimum distance between the spaces. If the two-sided equation has nontrivial solutions, the obtained vectors present a solution. Otherwise, these vectors compose a pseudo-solution that minimizes the deviation between both sides of the equation.