Understand the idea.
Define what best means
The objective function measures what the decision should improve. Specify whether to maximise profit, minimise cost or meet another stated aim. Constraints describe what is allowed; they do not by themselves identify the aim.
For profit, subtract each unit cost from its selling price before multiplying by quantity. A fixed cost reduces every plan by the same amount, so it does not change the best feasible mix when the constraints stay the same.
A family of parallel lines
Each equation with a fixed value of describes a line. Changing moves that line parallel to itself. This gives a graphical way to see which feasible boundary is reached by better objective values.
On a non-empty bounded polygon, a linear objective reaches an optimum at a vertex. Compare every feasible vertex. If the objective is constant along a whole feasible edge, all points on that edge tie; an optimum need not be unique.
See the method in action.
Profit rather than revenue
Basic kits sell for AUD 18 and cost AUD 11. Deluxe kits sell for AUD 26 and cost AUD 15. Write profit for x basic and y deluxe kits; evaluate at .
Worked solution
- 1
Calculate the profit per kit.
Selling prices include the costs that must first be recovered.
- 2
Write the total profit objective.
Each basic kit contributes AUD 7 and each deluxe kit contributes AUD 11.
- 3
Evaluate the proposed mix.
Six basic and four deluxe kits produce AUD 86 profit under the stated unit costs.
; AUD 86.
Compare corners
A region has vertices . Maximise .
| Vertex | P |
|---|---|
| (0,0) | 0 |
| (5,0) | 20 |
| (3,4) | 32 |
| (0,6) | 30 |
Worked solution
- 1
Evaluate the first two corners.
Substitute each feasible corner into the same objective .
- 2
Evaluate the remaining corners.
All four corner values are needed for the comparison.
- 3
Select the largest value.
A linear objective on the given bounded polygon attains a maximum at a vertex; has the largest value.
Maximum 32 at .
Try it yourself.
Check your understanding
Write profit when each x earns AUD 9 and each y earns AUD 4. Evaluate at .
Show answer & reasoning
; AUD 46.
The profit objective is . At it gives , so the profit is AUD 46.
Keep practising
6 worksheet questions · 12 marks
Read the questions online
- A has unit revenue 35 and cost 23; B has revenue 28 and cost 19. Write profit.2 marks
- Evaluate at .2 marks
- At vertices , maximise .2 marks
- At vertices , minimise .2 marks
- Does subtracting a fixed AUD 100 change the maximiser of ?2 marks
- Why does every point from to have the same ?2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- A has unit revenue 35 and cost 23; B has revenue 28 and cost 19. Write profit.2 marks
Subtract each unit cost from revenue first.
. - Evaluate at .2 marks
.
71. - At vertices , maximise .2 marks
The values are 0,24,32,28.
Maximum 32 at . - At vertices , minimise .2 marks
The values are 16,41,32.
Minimum 16 at . - Does subtracting a fixed AUD 100 change the maximiser of ?2 marks
All objective values decrease equally, preserving their order.
No, if the feasible set stays unchanged. - Why does every point from to have the same ?2 marks
On that edge , so .
Each has .
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 2 General Mathematics: Modelling with Linear Relationships, a compulsory school-assessed topic. The external examination scope differs from the full course.
Official curriculum source ↗