Solving an LPP by the Graphical (Corner-Point) Method

Once an LPP has been formulated and its feasible region graphed, the graphical method (also called the corner-point method) finds the optimal value of the objective function using a single remarkable fact, stated here without proof — the syllabus's own scope calls for the graphical method only, not a proof of why it works:

This is precisely why the method never needs to test every one of the infinitely many points inside the feasible region — only the handful of corner points, which can be found exactly by ordinary algebra.

The procedure, step by step:

1. Graph every constraint's boundary line and shade to find the feasible region (as in the previous two sections).
2. List every corner point of the feasible region — every point where two boundary lines meet, found by solving the corresponding pair of boundary equations simultaneously, PLUS the points where a boundary line meets an axis (found by setting $x=0$ or $y=0$ in turn). A point counts as a genuine corner of the feasible region only if it also satisfies every OTHER constraint of the problem — an intersection that violates some other constraint is not a corner point of the feasible region and must be discarded.
3. Evaluate $Z$ at every corner point found in step 2.
4. The largest of these values is the maximum of $Z$ on the feasible region, and the smallest is the minimum — read off whichever the problem actually asks for. …