Solve graphically: Maximise Z = 3x + 2y subject to 2x + y ≤ 10, x + y ≤ 6, x + 2y ≤ 10, x ≥ 0, y ≥ 0.
How to lay out the answer. Follow the frame, then write it in your own words.
1 · Write what is given and what is to be found. Given: Z = 3x + 2y to maximise, three non-trivial constraints and x, y ≥ 0. To find: the maximum of Z and where it occurs.
2 · Draw the figure or sketch.
2x + y = 10 through (5, 0) and (0, 10)
x + y = 6 through (6, 0) and (0, 6)
x + 2y = 10 through (10, 0) and (0, 5)
O satisfies all three: shade towards O and name the region OABCD
3 · Write the theorem. Corner point theorem: on a bounded region, the maximum of Z occurs at a corner point.
4 · Find the corner points.
A(5, 0); B(4, 2) from 2x + y = 10 and x + y = 6
C(2, 4) from x + y = 6 and x + 2y = 10; D(0, 5)
Each corner satisfies all three constraints
5 · Make the table of Z.
O(0, 0): Z = 0
A(5, 0): Z = 15
B(4, 2): Z = 12 + 4 = 16
C(2, 4): Z = 6 + 8 = 14
D(0, 5): Z = 10
6 · Check and box the answer. The region is bounded, so the largest corner value is the maximum. Maximum Z = 16 at x = 4, y = 2.
Scoring tip: Draw the table of Z values even when the answer looks clear from the graph. It carries a mark.
