Try these six questions on paper first, then open each answer. They run through the skills in the linear programming chapter in the order a full question uses them.
Questions
Question 1. A hall sells x adult tickets and y child tickets.
The number of child tickets is at least three times the number of adult tickets, and the total is at most 120. Write the inequalities.
Answer
Children at least three times adults: y ≥ 3x. Total: x + y ≤ 120. Neither number can be negative: x ≥ 0 and y ≥ 0.
Trial check: x = 10, y = 40. Then 40 ≥ 30 and 50 ≤ 120, so the point is allowed.
Question 2. For the region x + y ≤ 7, x ≥ 1 and y ≥ 1, find the maximum of P = 2x + 5y.
Answer
Vertices: (1, 1), (6, 1) and (1, 6).
P values: (1, 1) gives 7. (6, 1) gives 17. (1, 6) gives 32.
The maximum is 32 at (1, 6), because 2(1) + 5(6) = 2 + 30.
Question 3. Draw the region x + y ≤ 8 and y ≥ 2 with x ≥ 0. Give its vertices.
Answer
x + y = 8 is solid, with the origin side kept. y = 2 is solid, with the upper side kept.
Vertices: y = 2 and x = 0 give (0, 2). y = 2 and x + y = 8 give (6, 2). x = 0 and x + y = 8 give (0, 8).
The region is a triangle with those three vertices.
Question 4. In the region with vertices (1, 1), (6, 1) and (1, 6), the profit is P = x + y. Find the maximum and describe where it occurs.
Answer
P values: (1, 1) gives 2. (6, 1) gives 7. (1, 6) gives 7.
The maximum is 7. It occurs at both (6, 1) and (1, 6), and so at every point on the edge x + y = 7 between them.
Question 5. A shop sells x bouquets and y potted plants.
It sells at most 18 items in total and at least twice as many bouquets as plants. Profit is RM5 per bouquet and RM8 per plant. Find the maximum profit and say what the shop should sell.
Answer
Constraints: x + y ≤ 18, x ≥ 2y, with x ≥ 0 and y ≥ 0.
Vertices: (0, 0), (18, 0), and x = 2y with x + y = 18, where 3y = 18, giving y = 6 and x = 12, so (12, 6).
P = 5x + 8y. Values: (0, 0) gives 0. (18, 0) gives 90. (12, 6) gives 60 + 48 = 108.
“The shop should sell 12 bouquets and 6 potted plants for a maximum profit of RM108.”
Question 6. A student finds the vertex (4, 9) for the constraints x + y ≤ 10 and y ≥ x + 2. Explain whether it is a vertex.
Answer
Substitute (4, 9) into x + y ≤ 10: 13 ≤ 10 is false. The point breaks the first constraint, so it cannot be a vertex of the feasible region.
The true vertex on y = x + 2 and x + y = 10 comes from 2x + 2 = 10, so x = 4 and y = 6, giving (4, 6).
If you got these wrong
- Question 1 tests writing constraints from a word problem.
- Questions 3 and 6 test drawing the feasible region.
- Questions 2 and 4 test testing an objective function at vertices.
- Question 5 needs explaining a maximum or minimum in context.
Use the mistake log to note which step failed, and the timed original practice session builder for a timed set. For a teacher to review your working, see online one-to-one Additional Mathematics tuition.