The graphical method
Draw each constraint as a line and shade the side that satisfies its inequality. The region satisfying all constraints at once is the feasible region. Its corners are the vertices, and each one is where two boundary lines cross. Because the feasible region is bounded by straight lines, it always has the shape of a polygon.
Optimum at a vertex
A key result: the maximum or minimum of a linear objective function over a feasible region always occurs at a vertex. So you only need to evaluate the objective at each corner and compare — no need to test every interior point.
In practice you draw each boundary line, shade the unwanted region for every inequality so that the feasible region is left clear, and then read off the coordinates of each corner. Where two boundary lines meet, solve them as simultaneous equations to get the exact vertex. If the variables must be whole numbers, also check the integer points nearest the best vertex, since the true optimum may not sit exactly on a corner.
Key formula /
To optimise P = ax + by: list the vertices of the feasible region, compute P at each, then choose the largest value (for a maximum) or smallest (for a minimum).
Worked example
A feasible region has vertices (0, 0), (6, 0), (0, 4) and (4, 2). Maximise P = 3x + 2y. Evaluate: (0,0) → 0; (6,0) → 18; (0,4) → 8; (4,2) → 12 + 4 = 16. The largest is 18 at (6, 0), so the maximum profit is 18.
Remember
- The optimum is always at a vertex.
- Read vertices from the graph carefully.
- Compare all corner values before deciding.