# Graphically solving a Linear Programming Problem?

I was given the following linear programming problem and have been asked to find all optimal solutions graphically. I am quite new to the subject, so please forgive my naivety.

minimize:
z = x - y
subject to:
x + y <= 6
x - y >= 0
y - x >= 3
x, y >= 0


I've graphed the constraints as shown below: To me, it seems like there are no optimal solutions because there is no feasible region (that is, a region where all constraints are satisfied and therefore all three shaded colors would overlap). Is my assumption correct or is there an optimal solution(s) to this problem and how can I go about finding it(them) graphically?

• If you add the two constraints $x-y\geq0$ and $y-x\geq 3$, you get $0\geq 3$, which cannot be. So, indeed, either the constraint set is empty and thus there is no solution, or there is a typo somewhere. – triple_sec Sep 9 '13 at 1:13
• True, I never thought about that. But I'm pretty sure there are no typos here, so there must be no solution. – audiFanatic Sep 9 '13 at 1:17
• Please make your answer an answer to the question so I may properly give you credit. – audiFanatic Sep 9 '13 at 1:21
• @audiFanatic may I know how have you plotted such a colorful and accurate graph, any software or website? – Dutta Sep 9 '13 at 1:42
• I used a freeware program plainly called "Graph." Very simple and easy to use: padowan.dk/download – audiFanatic Sep 9 '13 at 1:45