Q.Maximize : $z = 7x + 11y$ subject to $3x + 5y \le 26,\ 5x + 3y \le 30,\ x \ge 0,\ y \ge 0$.

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 found; (5) the largest tabulated value of $z$ is the maximum, and the smallest is the minimum, of the objective function over the entire feasible region. If two different vertices happen to give the SAME optimal value of $z$, then (by the same convexity idea) every point on the straight segment joining those two vertices is also optimal, so the L.P.P. has infinitely many optimal solutions rather than a single unique one.