The last line of a linear programming answer must say what the best vertex means in the situation. A coordinate alone, such as (9, 3), does not answer a question that asks “how many of each should be made”.
This lesson finishes the skills in the linear programming chapter. Before it, you need the region and the values from testing an objective function at vertices.
What should a complete answer contain?
Include three things: the decision (how many of each), the value (with units) and the word maximum or minimum. If the question adds a follow-up, such as a target amount, answer that in a separate sentence.
Worked example: charity food sale
A charity sells x plain packs and y special packs of nasi lemak. It prepares at most 12 packs in total, so x + y ≤ 12. It must sell at least 2 plain packs and at least 3 special packs: x ≥ 2 and y ≥ 3. The profit is 6 per plain pack and 4 per special pack, so P = 6x + 4y (in RM).
The vertices of the region are (2, 3), (9, 3) and (2, 10). Test each one:
| Vertex (x, y) | 6x + 4y | P (RM) |
|---|---|---|
| (2, 3) | 12 + 12 | 24 |
| (9, 3) | 54 + 12 | 66 |
| (2, 10) | 12 + 40 | 52 |
The maximum is at (9, 3). The complete answer reads:
“The charity should sell 9 plain packs and 3 special packs to earn the maximum profit of RM66.”
Which limits are fully used?
The vertex (9, 3) lies on y = 3 and x + y = 12. So the limit of at least 3 special packs is met exactly, and all 12 packs are used. The plain-pack minimum of 2 is exceeded.
This is useful because it explains why the answer is where it is. Profit per plain pack is higher, so the charity uses its spare capacity on plain packs and only the minimum number of special packs.
A what-if question
The question might ask: “Can the charity earn RM70?” Since the maximum is RM66, it cannot. The reason is that RM70 is larger than the maximum profit over the whole feasible region.
Answer with the comparison: “No, because the maximum profit is RM66, which is less than RM70.”
The mistake that costs marks
Compare two endings for the same working:
| Ending | Wrong | Right |
|---|---|---|
| Answer | (9, 3) | 9 plain packs and 3 special packs |
| Value | 66 | Maximum profit RM66 |
| Type | not stated | Maximum stated |
The second version names the items, gives the unit and says what the number is. Use the question’s vocabulary, not x and y.
Check yourself
A baker bakes x trays of muffins and y trays of tarts with constraints x + y ≤ 10, x ≤ 6 and y ≥ 2. Profit per tray is RM40 for muffins and RM30 for tarts. The vertices are (0, 2), (6, 2), (6, 4) and (0, 10). Write the full conclusion for the maximum profit.
Answer
P = 40x + 30y.
Values: (0, 2) gives 60, (6, 2) gives 300, (6, 4) gives 360 and (0, 10) gives 300.
The maximum is at (6, 4).
“The baker should bake 6 trays of muffins and 4 trays of tarts for a maximum profit of RM360.”
The vertex lies on x = 6 and x + y = 10, so the muffin limit and the total tray limit are used up.
What to study next
Practise the whole chain on the linear programming practice set. Then log any lost marks in the mistake log.
If you would like a teacher to shape your written conclusions with you, see online one-to-one Additional Mathematics tuition.