Most of the time the answer is not unique since the optimal solution is usually a vertex of the simplex in the solution space and you can have infinitely many hyperplanes which admits extrema at that point. The only time that the answer could be unique is that the hyperplane the objective function make happens to be parallel to one of the face of the simplex, and the answer you produce is in the strict interior of that face (if it's vertex, or somewhere on the edge, then still infinite possible objective functions)