As I am sure you are aware, a system of linear equations has (a) no valid solution, (b) a unique valid solution, or (c) an infinite number of valid solutions. The latter possibility, however, is less probable than either of the other two.

A system of linear inequations (or a mixture of linear equations and inequations) has the same three possibilities, but now the probability of a unique solution is smaller than the probability of either of the other two. Linear programming finds a solution (assuming one exists) that optimizes some objective function. You did not say whether your problem said thatAsolution was

x_{1}= 0.05, x_{2}= 0.3, and x_{3}= 0.65, implying that the solution was not unique, or whether your problem said that it wasTHEsolution, implying a unique solution. Now if the solution given is the only solution to the system of given equations and inequations, it presumably could be found using linear programming. (This is a guess on my part because I do not know what happens to the simplex method when the feasible region consists of a single point. I am assuming the simplex method does not go insane in that case.) If the solution given is the only solution, but the system itself has an infinite number of solutions, then there must be an objective function hidden somewhere in the statement of the problem, and the answer can be found using linear programming. If the solution given is not unique, then one way to find one of those solutions is to provide an arbitrary objective function and then apply linear programming. Different objective functions would give different solutions of course, but if the goal is to find any valid answer, you do not care which valid solution you get. See what happens if you give an objective function of O(x_{1}, x_{2}, x_{3}) = x_{3}and treat the system of equations and inequations as constraints in a linear programming calculator.

Edit: you may need to play around making sure that the non-negativity constraints work.