Every linear-programming optimum lives on a corner. Build the feasible region from up to six inequality constraints, watch the objective line slide across it, and read the winning vertex straight off the table — with unbounded and infeasible cases diagnosed instead of silently ignored.
| x | y | z |
|---|
The optimum of any linear objective over a polygon is always at a corner (or along the edge between two equal corners — highlighted when that happens). “≤” regions are shaded below their line; the tinted intersection is what satisfies everything at once.