Linear Programming for IGCSE Maths

What Is Linear Programming in IGCSE Maths?

Linear programming is a topic on the IGCSE Extended syllabus that combines inequality graphing with optimisation. It typically appears as a longer question worth six to eight marks on Paper 2 or Paper 4. Despite its slightly intimidating name, linear programming follows a clear, step-by-step method that is highly repeatable.

The basic idea is this: you have several constraints (written as inequalities), you graph them to find a feasible region, and then you find the point within that region that maximises or minimises a given expression. It is one of the few IGCSE topics that has direct real-world applications — businesses use linear programming to maximise profit and minimise costs every day.

Step 1: Write the Inequalities

The question will describe a real-world scenario with constraints. Your first job is to translate these into mathematical inequalities.

For example, a bakery makes cakes (x) and pastries (y) with these constraints:

Each cake needs 2 kg of flour; each pastry needs 1 kg. The bakery has 20 kg of flour. This gives 2x + y ≤ 20.
Each cake needs 1 hour to bake; each pastry needs 1.5 hours. The oven is available for 15 hours. This gives x + 1.5y ≤ 15.
You cannot make a negative number of cakes or pastries: x ≥ 0 and y ≥ 0.

Always include the non-negativity constraints (x ≥ 0 and y ≥ 0) unless the question specifies otherwise.

Step 2: Graph Each Inequality

For each inequality, draw the boundary line and shade the correct region:

To draw 2x + y = 20: find two points. When x = 0, y = 20. When y = 0, x = 10. Plot these points and draw a straight line.
To draw x + 1.5y = 15: when x = 0, y = 10. When y = 0, x = 15. Plot and draw.

For each line, decide which side satisfies the inequality. The easiest way is to test the point (0, 0):

For 2x + y ≤ 20: is 0 + 0 ≤ 20? Yes, so shade the side that includes the origin.
For x + 1.5y ≤ 15: is 0 + 0 ≤ 15? Yes, so shade the side that includes the origin.

Be careful with the shading convention. Some questions ask you to shade the unwanted region, leaving the feasible region clear. Others ask you to shade the feasible region. Read the instruction carefully.

Step 3: Identify the Feasible Region

The feasible region is the area where all inequalities are satisfied simultaneously. It is the overlap of all the individual regions. On your graph, this should be a clearly defined polygon.

Mark the vertices (corner points) of the feasible region. These are the points where the boundary lines intersect. You can find exact coordinates by solving pairs of simultaneous equations.

In our bakery example, the vertices might be at (0, 0), (10, 0), (0, 10), and the intersection of 2x + y = 20 and x + 1.5y = 15.

To find that intersection, solve simultaneously:

From 2x + y = 20: y = 20 - 2x
Substitute into x + 1.5(20 - 2x) = 15: x + 30 - 3x = 15, so -2x = -15, giving x = 7.5 and y = 5.

Step 4: Find the Optimal Solution

The question will give you an objective function to maximise or minimise. For example, “the profit is P = 5x + 3y. Find the maximum profit.”

The optimal solution always occurs at a vertex of the feasible region. Test each vertex:

At (0, 0): P = 0
At (10, 0): P = 50
At (7.5, 5): P = 37.5 + 15 = 52.5
At (0, 10): P = 30

The maximum profit is 52.5 at the point (7.5, 5).

If the question requires whole-number (integer) solutions, check the integer points near the optimal vertex and pick the best one.

Important Graphing Tips

Accuracy is essential in linear programming questions. Follow these tips:

Use a sharp pencil and ruler. Freehand lines lose marks.
Choose a sensible scale. Make sure all the boundary lines fit on the graph paper. Look at the intercepts to decide the range of your axes.
Label each line with its equation so the examiner can identify them.
Use different types of shading (e.g., diagonal lines in different directions) for different inequalities. This makes it easy to see the overlap.
Mark the feasible region clearly with an R or label it “feasible region.”

Solid vs Dashed Lines

This detail is easy to forget but carries marks:

≤ or ≥ uses a solid line (the boundary is included in the region).
< or > uses a dashed line (the boundary is not included).

Most IGCSE linear programming questions use ≤ and ≥, so solid lines are more common. But always check.

Common Mistakes to Avoid

Shading the wrong side of a line. Always test a point (usually the origin) to confirm.
Not finding exact intersection points. Reading approximate coordinates off the graph loses accuracy marks. Solve the simultaneous equations algebraically.
Forgetting the non-negativity constraints. The feasible region must be in the first quadrant (where x ≥ 0 and y ≥ 0).
Testing only some vertices. You must evaluate the objective function at every vertex to be sure you have found the true optimum.
Ignoring integer constraints. If the question says “x and y are whole numbers,” the optimal vertex might not be valid, and you need to check nearby integer points.

Practice Questions

Graph the region defined by x + y ≤ 10, 2x + y ≤ 16, x ≥ 0, y ≥ 0.
Find the vertices of the feasible region.
Maximise P = 3x + 4y subject to these constraints.
If x and y must be integers, what is the maximum value of P?

Summary

Linear programming is a structured topic with a clear four-step method: write inequalities, graph them, identify the feasible region, and test the vertices. Accuracy in graphing and algebra is essential. Once you have practised the method a few times, this becomes a reliable source of marks on the Extended paper.

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