Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who need to graph inequalities, shade feasible regions and find maximum or minimum values of an objective function.
What query it owns: how to solve linear programming problems in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the revision-guide angle, while Tutopiya’s Linear Programming subtopic page owns the learning resource and the free Linear Programming quiz owns the practice.

Linear programming is the bridge between algebra and real-world optimisation in Cambridge IGCSE Mathematics (0580/0607). You graph linear inequalities to find a feasible region, then evaluate an objective function at the corner points to find the best value. This guide walks through the method step by step and shows how to answer the command words examiners use.

Key takeaways

• Linear programming finds the maximum or minimum of an objective function subject to linear constraints.
• Graph each inequality as a boundary line; use a test point to decide which side to shade.
• The optimal value occurs at a corner (vertex) of the feasible region.
• Inequality symbols matter: ≤ and ≥ include the boundary; < and > use a dashed line.

What is linear programming in Cambridge IGCSE Maths?

Linear programming solves optimisation problems where the constraints and objective are linear. Typical contexts include maximising profit or minimising cost subject to limits on materials or time. You represent constraints as inequalities in x and y, shade the region that satisfies all of them (the feasible region), and test the objective function P = ax + by at each corner point.

Read the full notes on Tutopiya’s Linear Programming subtopic page before you attempt questions.

The core ideas you must master

Idea	What it means	Exam signal
Constraint	A limit written as a linear inequality	”At most 100 units…”
Feasible region	Area satisfying all constraints	”Shade the region R.”
Objective function	Expression to maximise or minimise	”P = 3x + 2y”
Corner point	Vertex of the feasible region	Where optimal value occurs

How to solve a linear programming problem — step by step

1. Define variables (e.g. x = number of items A, y = number of items B).
2. Write constraints as inequalities including x ≥ 0, y ≥ 0 if needed.
3. Draw each boundary line — solid for ≤/≥, dashed for </>.
4. Shade the correct side for each inequality; the overlap is the feasible region.
5. Find corner points by solving pairs of boundary equations.
6. Evaluate the objective function at each corner; pick the max or min as required.

Test the method with the free Linear Programming quiz.

Graphing inequalities: solid, dashed and shading

Symbol	Boundary line	Region
y ≤ mx + c	Solid	Below the line
y ≥ mx + c	Solid	Above the line
y < mx + c	Dashed	Below the line
y > mx + c	Dashed	Above the line

Use (0, 0) as a test point when it is not on the line: substitute into the inequality; if true, shade that side.

Linear programming in past-paper wording: command words that matter

Command word / phrase	What the question wants	Typical stem
Draw the line	Accurate boundary on axes	”Draw the line 2x + y = 10.”
Shade the region	Feasible area labelled R	”Shade the region R where x ≥ 0, y ≥ 0 and x + y ≤ 8.”
Find the maximum value	Evaluate objective at corners	”Find the maximum value of P = 4x + 3y.”
Write down the inequalities	Translate a word problem	”Write down the inequalities that represent the constraints.”
Use your graph	Read values from the feasible region	”Use your graph to find the minimum value of C.”

Worked exam-style stems (how to answer the wording)

1. “Shade the region R where x ≥ 0, y ≥ 0, x + y ≤ 6 and 2x + y ≤ 8.” Draw four boundaries; shade the overlap in the first quadrant. Corner points include (0,0), (0,6), (2,4), (4,0). Reward: correct shading, region labelled R.

2. “P = 5x + 2y. Find the maximum value of P for points in region R.” Test P at each corner of R. Largest value wins — e.g. if corners give P = 0, 12, 18, 20, maximum is 20. Reward: all corners tested, maximum stated.

3. “A factory makes chairs (x) and tables (y). It makes at most 50 items in total. Write down an inequality.” x + y ≤ 50 (plus x ≥ 0, y ≥ 0). Reward: correct inequality with context.

When you can recognise the wording instantly, work the full set on the Coordinate Geometry topical past paper questions and the Linear Programming quiz.

How linear programming connects to Coordinate Geometry

Linear programming depends on Equation of a Line for drawing boundaries and Distance, Midpoint and Gradient for reading graphs accurately. Use the Cambridge IGCSE Maths resource hub to revise weak areas.

Common mistakes students make

• Shading the wrong side of an inequality.
• Using a dashed line when the inequality includes equals (≤ or ≥).
• Forgetting x ≥ 0, y ≥ 0 in the first quadrant.
• Testing the objective at only one corner instead of all vertices.

When you need more support

If feasible regions or objective functions keep costing marks, work through the Equation of a Line quiz and the Coordinate Geometry topical past papers, then get help from a Cambridge IGCSE Maths tutor.

Frequently asked questions

What is a feasible region in linear programming? It is the set of points (x, y) that satisfy every constraint inequality simultaneously — the shaded overlap on your graph.

Where does the maximum or minimum occur? At a corner point (vertex) of the feasible region. Evaluate the objective function at every corner and compare.

Do I need to shade accurately in the exam? Yes — clear boundary lines, correct shading and a labelled region R are worth method marks even before you find the optimum.

How should I revise linear programming? Read the subtopic notes, work two full problems including shading, then take the Linear Programming quiz.

Ready to master Cambridge IGCSE Maths linear programming?

Start with the Linear Programming subtopic page, then book a free trial with a Cambridge IGCSE Maths specialist to turn optimisation questions into guaranteed marks.

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