Graphical method

The essentials

A two-variable LPP can be solved by drawing. The method has four steps.

• Draw each constraint as a straight line (replace the inequality by "=") using its intercepts, and shade the side that satisfies the inequality. Testing the origin (0, 0) tells you which side to shade, unless the line passes through it.
• The region common to all the constraints, including x ≥ 0 and y ≥ 0, is the feasible region.
• Find the corner points (vertices) of the feasible region, solving pairs of line equations where needed.
• Evaluate Z at each corner point. For a bounded region, the largest value is the maximum and the smallest is the minimum.

This works because of the corner point theorem: if an optimal value exists, it occurs at a corner point of the feasible region.

Key points

• If Z takes the same optimal value at two adjacent corners, every point on the segment joining them is also optimal.
• Example: maximise Z = 3x + 4y subject to 2x + y ≤ 10, x + 3y ≤ 15, x, y ≥ 0. The corners are (0, 0), (5, 0), (3, 4) and (0, 5), with Z = 0, 15, 25 and 20, so the maximum is 25 at (3, 4).
• The point (3, 4) comes from solving 2x + y = 10 and x + 3y = 15 together.

Exam tips

• Make a corner-point table with columns for the point and the value of Z; it organises the work and earns method marks.
• Draw on a sensible scale and label each line.

Common mistakes

• Missing a corner formed by two constraint lines (not just axis intercepts).
• Shading the wrong side of a ≥ constraint.