Q.Minimize $z = 4x + 5y$ subject to $2x + y \ge 7$, $2x + 3y \le 15$, $x \le 3$, $x \ge 0$, $y \ge 0$. Solve using graphical method.

The chapter proves (without proof, as a stated theorem) that when the objective function of an L.P.P. attains its optimum (maximum or minimum) value over a convex-polygon feasible region, that optimum is always attained at one of the polygon's vertices (corner points) — this is the Corner-Point or Convex-Polygon Theorem, and it is what makes the graphical method possible: instead of checking every point of an infinite region, only the (finitely many) vertices need to be checked. The Corner-Point Method proceeds in five steps: (1) convert every constraint's inequation into its boundary equation and draw the line; (2) shade the half-plane each constraint demands and locate the region common to all of them — the feasible region; (3) find every vertex of this feasible region, either by reading intercepts directly off the graph or, more reliably, by solving each relevant pair of boundary-line equations simultaneously; (4) evaluate the objective function $z$ at every vertex f …