Linear programming finds the best value of a quantity, such as profit or cost, when a situation has limits. A question gives you a story, and your job is to turn the story into inequalities, a region on a graph and a final sentence.
This chapter sits inside the SPM Additional Mathematics guide. The skills build in a fixed order, and each one has its own lesson.
How do the four skills fit together?
Every linear programming question follows the same chain. A mistake early in the chain carries through, so the order is the study order too.
- Turning a situation into linear constraints: read the words and write the inequalities.
- Drawing the feasible region: draw the lines and shade the region that satisfies all of them.
- Testing an objective function at vertices: calculate profit or cost at each corner.
- Explaining a maximum or minimum in context: write what the answer means in the situation.
A short orienting example
A stall sells x bowls of soup and y plates of noodles in one evening. It has room for at most 30 items in total, so x + y ≤ 30. The stall sells at least 5 bowls of soup, so x ≥ 5.
On a graph, x + y = 30 is a slanted line and x = 5 is a vertical line. The region meeting both limits, with x and y not below zero, is a triangle.
If profit is 3 per bowl and 4 per plate, the profit 3x + 4y is calculated at the triangle’s corners. The largest value marks the best plan. You will meet this full chain again in the lessons, with more limits and a written conclusion.
Who should start where?
Match your situation to a starting point:
- If sentences like “at least twice as many” confuse you, begin with constraints.
- If your inequalities are right but the shading is unreliable, begin with the feasible region.
- If you draw well but lose marks on the final answer, go to testing vertices and explaining the result.
- If you want to test yourself, use the linear programming practice set.
The word-problem structure worksheet helps with reading long question stems into parts. For a teacher who can work through your own mistakes, see online one-to-one Additional Mathematics tuition.