Finding basic feasible solution graphically

Question

I'm supposed to find the basic solutions of the given LPP graphically. I know what a bfs means, I can find that in other way (I mean by method mentioned in this post), but I don't know how to do it graphically. The system is:- $$2x + 3y \leq 21$$$$ 3x - y \leq 15$$$$ x + y \geq 5$$$$ y \leq 5$$ $$ x, y \geq 0$$ Any hint/solution would be appreciated. Thanks!

Answer 1

For a linear program like this, the set of all feasible solutions is the intersection of half spaces and thus a polyhedron. A BFS corresponds to a vertex of this polyhedron.

Now, since we only have 2 variables we can plot all our half spaces in the plane and take the intersection to get a polygon. Then, the BFS are just the vertices of this polygon.

Here is a graph of your problem (more or less)

From that graph we can see (0,5), (5,0), (3,5), and (6,3) are all BFS.