What is Linear Programming and how is it used in Coordinate Geometry?

Linear Programming is a mathematical method used to determine the best possible outcome in a given mathematical model. Its functions are linear, and it involves constraints that are also linear equations. In Coordinate Geometry, Linear Programming is used to find the optimal solution by graphing inequalities on a coordinate plane and identifying feasible regions where these inequalities overlap.

How do you formulate a Linear Programming problem in Mathematics?

To formulate a Linear Programming problem, you need to define the objective function, which is the function you want to maximize or minimize. Next, identify the constraints, which are the linear inequalities that limit the values of the variables. Finally, graph these inequalities on a coordinate plane to find the feasible region, which is the set of all possible solutions that satisfy the constraints.

What are the steps to solve a Linear Programming problem graphically?

To solve a Linear Programming problem graphically, follow these steps: 1) Define the objective function and constraints. 2) Graph the constraints on a coordinate plane to form a feasible region. 3) Identify the vertices of the feasible region. 4) Evaluate the objective function at each vertex. 5) Choose the vertex that gives the optimal value of the objective function, either maximum or minimum, depending on the problem.

What is the significance of the feasible region in Linear Programming?

The feasible region in Linear Programming is the area on a graph where all the constraints of the problem overlap. It represents all possible solutions that satisfy the constraints. The significance of the feasible region is that the optimal solution to the Linear Programming problem, whether maximizing or minimizing the objective function, will always be found at one of the vertices of this region.

Can you explain the role of constraints in a Linear Programming problem?

Constraints in a Linear Programming problem are the linear inequalities that define the limits within which the solution must lie. They are essential because they shape the feasible region on the coordinate plane. The constraints ensure that the solution is practical and adheres to the conditions of the problem. Without constraints, the objective function could be optimized without any real-world applicability.

Why is "Using a solid line for a strict inequality (or vice versa)" a common mistake in Linear Programming?

Why it happens: Forgetting which inequality type uses which line. How to avoid it: → dashed (boundary excluded). ≤, ≥ → solid (boundary included).

Why is "Shading the wrong side of a boundary" a common mistake in Linear Programming?

Why it happens: Confusion about which side the inequality represents. How to avoid it: Test (0, 0) in the inequality. If true, (0,0) is in the region — shade that side. Source: 0580/42 — recurring

Why is "Picking a non-integer optimum when the question requires integers" a common mistake in Linear Programming?

Why it happens: Continuous LP gives the optimum at a vertex; integer LP may need a slightly different point. How to avoid it: If x, y must be whole numbers, check INTEGER points near the continuous optimum.

Why is "Forgetting $x \geq 0,\ y \geq 0$" a common mistake in Linear Programming?

Why it happens: Students focus on the worded constraints and miss the implicit non-negativity. How to avoid it: When x or y counts physical objects, non-negativity is automatic — write it.

Coordinate GeometryCambridge IGCSE Mathematics (0580)

Linear Programming

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Linear Programming — Cambridge IGCSE 0580 Maths Extended (2026)

Plot inequalities, identify the feasible region, and find the optimum value of an objective. The skill that combines coordinate geometry, inequalities and a sketch of practical thinking — recurring on Paper 4.

What you’ll learn

Mapped to the Cambridge IGCSE 0580 syllabus (2025-2027).

E3.4 — Solve simple linear programming problems graphically; represent constraints by inequalities and identify feasible regions.

Plotting an inequality

Treat the inequality as an equation, plot the line, decide solid vs dashed, then shade.

Method.

Replace the inequality sign with $=$ to get a boundary line.
Plot the line:
- Dashed if STRICT ( $<$ or $>$ ).
- Solid if INCLUSIVE ( $\le$ or $\ge$ ).
Test a point NOT on the line (the origin if possible). If it satisfies the inequality, shade THAT side. Otherwise shade the OPPOSITE side.

Worked. Plot $y \ge 2x + 1$ .

Boundary: $y = 2x + 1$ . Solid line.
Test origin $(0, 0)$ : $0 \ge 1$ ? No. So shade the side AWAY from the origin (the upper side).

Worked. Plot $x + y < 5$ .

Boundary: $x + y = 5$ . Dashed.
Test origin: $0 < 5$ ? Yes. Shade the side CONTAINING the origin (lower-left).

Cambridge usually wants you to shade the region that satisfies the inequality (the WANTED region). Some textbooks shade the UNWANTED region — read the question carefully.

Replace $\le, <, \ge, >$ with $=$ to draw the line.
Solid for $\le, \ge$ . Dashed for $<, >$ .
Test a point off the line.
Shade the side satisfying the inequality (default Cambridge convention).

Feasible region

The overlap of all the shaded regions. Where ALL constraints are satisfied at once.

When a problem has multiple inequalities, plot each one and find the OVERLAP. That overlap is the feasible region — the set of points satisfying ALL constraints.

Worked. Find the feasible region for:

$x + y \le 8$ .
$y \ge 2$ .
$x \ge 0$ .

Plot each:

$x + y = 8$ (solid). Test origin: $0 \le 8$ ✓. Shade below.
$y = 2$ (solid). Shade above.
$x = 0$ is the $y$ -axis. Shade to the right.

The feasible region is the overlap of those three shaded areas — a triangle.

Vertices. The corners of the feasible region. They lie at intersections of boundary lines:

$y = 2$ and $x = 0$ → $(0, 2)$ .
$x + y = 8$ and $y = 2$ → $(6, 2)$ .
$x + y = 8$ and $x = 0$ → $(0, 8)$ .

Plot each inequality.
Identify the OVERLAP — feasible region.
Find vertices at boundary intersections.
List all vertices in case the optimum is at one of them.

Optimising an objective

The maximum or minimum of a linear objective is always at a vertex of the feasible region.

Theorem. A linear objective function $z = ax + by + c$ takes its maximum and minimum on a feasible region at the VERTICES of that region (assuming the region is bounded).

Method.

List all vertices.
Substitute each vertex into the objective function.
The vertex giving the largest value is the MAXIMUM; the smallest is the MINIMUM.

Worked. From the previous example, find the maximum of $z = 3x + 2y$ .

Vertices: $(0, 2)$ , $(6, 2)$ , $(0, 8)$ .
$z(0, 2) = 4$ .
$z(6, 2) = 22$ .
$z(0, 8) = 16$ .
Maximum: $z = 22$ at $(6, 2)$ .

Word problems. Cambridge dresses linear programming inside resource-allocation contexts: "a factory produces $x$ tables and $y$ chairs; profit per table is …; constraints on materials are …; find the maximum profit." Translate the words into inequalities and an objective, then apply the method.

Optimum is at a VERTEX of the feasible region.
Substitute each vertex into the objective function.
Compare values to find max/min.
Word problems: translate constraints to inequalities, objective to $z = ax + by$ .

How it’s examined

Linear programming appears most years on Paper 4 as a 6-8 mark item. Cambridge gives a real-world scenario, expects you to translate to inequalities, sketch the feasible region, and find the optimum. Examiner reports flag missed dashed/solid distinction and wrong-side shading as recurring slips.

Sources: Cambridge IGCSE Mathematics 0580 syllabus 2025-2027 (E3.4); 0580/42 Oct/Nov 2024 — Q9 (linear programming); 0580 Examiner Reports 2022-2024. Last reviewed 2026-05-05.

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Step-by-step worked examples — Linear Programming

Step-by-step solutions to past-paper-style questions on linear programming, written exactly the way a tutor would explain them at the board.

1Translate constraints into inequalities
Extended• constraints
Question
A van carries $x$ small boxes and $y$ large boxes. There are at most $20$ boxes in total. Small boxes weigh $5$ kg, large weigh $12$ kg, and the total weight cannot exceed $156$ kg. Write the inequalities.
Step-by-step solution
1. Step 1
  Total boxes constraint.
  $x + y \leq 20$
2. Step 2
  Total weight constraint.
  $5x + 12y \leq 156$
3. Step 3
  Non-negativity.
  $x \geq 0,\ y \geq 0$
Answer
$x + y \leq 20,\ 5x + 12y \leq 156,\ x \geq 0,\ y \geq 0$
Examiner tip
Always include non-negativity constraints — boxes (or any physical count) can't be negative. Mark schemes credit them explicitly.
2Identify the feasible region
Extended• Adapted from 0580/42 May/Jun 2024 Q15• feasible region
Question
Sketch the region defined by $x + y \leq 8,\ y \geq x,\ x \geq 0$ .
Step-by-step solution
1. Step 1
  Draw $y = -x + 8$ (boundary of $x + y = 8$ ); shade BELOW.
2. Step 2
  Draw $y = x$ ; shade ABOVE.
3. Step 3
  $x \geq 0$ → shade right of the y-axis.
4. Step 4
  Feasible region is the intersection — a triangle with vertices $(0, 0)$ , $(0, 8)$ and $(4, 4)$ .
Answer
Triangular region with vertices $(0, 0)$ , $(0, 8)$ , $(4, 4)$ .
3Maximise an objective function
Extended• optimisation
Question
Subject to $x + y \leq 6$ , $x \leq 4$ , $y \leq 5$ , $x, y \geq 0$ , maximise $P = 3x + 4y$ over integer values.
Step-by-step solution
1. Step 1
  Find the corners: $(0, 0)$ , $(0, 5)$ , $(1, 5)$ , $(4, 2)$ , $(4, 0)$ .
2. Step 2
  Evaluate $P$ at each.
  $(0,0): 0;\ (0,5): 20;\ (1,5): 23;\ (4,2): 20;\ (4,0): 12.$
3. Step 3
  Maximum at $(1, 5)$ .
Answer
Maximum $P = 23$ at $(x, y) = (1, 5)$

Key Formulae — Linear Programming

The formulae you need to memorise for linear programming on the Cambridge IGCSE 0580 paper, with every variable defined in plain English and a note on when to use it.

Optimisation corner-point principle
$\text{Optimum of a linear } f \text{ on a feasible region occurs at a vertex (corner)}$
When to use
Linear-programming optimisation — only check the corner points of the feasible region.
Boundary line types
$\text{Strict (} <,> \text{)} \to \text{dashed line};\ \text{Non-strict (} \leq, \geq \text{)} \to \text{solid line}$
When to use
Drawing boundary lines for inequalities.

Key Definitions and Keywords — Linear Programming

Definitions to memorise and the exact keywords mark schemes credit for linear programming answers — sharpened from recent examiner reports for the 2026 0580 sitting.

Feasible region
Examiner keyword
The set of all points $(x, y)$ satisfying every constraint simultaneously.
Objective function
Examiner keyword
The expression to maximise or minimise (e.g. profit $P = 3x + 4y$ ).
Constraint
Examiner keyword
An inequality that limits the values $x$ and $y$ can take.
Vertex of feasible region
An intersection of two boundary lines that satisfies all constraints — a candidate for the optimum.

Common Mistakes and Misconceptions — Linear Programming

The traps other students keep falling into on linear programming questions — taken from recent Cambridge IGCSE 0580 examiner reports and mark schemes — and how to avoid them.

✕Using a solid line for a strict inequality (or vice versa)
Why it happens
Forgetting which inequality type uses which line.
How to avoid it
$<$ , $>$ → dashed (boundary excluded). $\leq$ , $\geq$ → solid (boundary included).
✕Shading the wrong side of a boundary
0580/42 — recurring
Why it happens
Confusion about which side the inequality represents.
How to avoid it
Test $(0, 0)$ in the inequality. If true, $(0,0)$ is in the region — shade that side.
✕Picking a non-integer optimum when the question requires integers
Why it happens
Continuous LP gives the optimum at a vertex; integer LP may need a slightly different point.
How to avoid it
If $x, y$ must be whole numbers, check INTEGER points near the continuous optimum.
✕Forgetting $x \geq 0,\ y \geq 0$
Why it happens
Students focus on the worded constraints and miss the implicit non-negativity.
How to avoid it
When $x$ or $y$ counts physical objects, non-negativity is automatic — write it.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

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Linear Programming — frequently asked questions

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Linear Programming

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Translate constraints into inequalities

2Identify the feasible region

3Maximise an objective function

Optimisation corner-point principle

Boundary line types

Feasible region

Objective function

Constraint

Vertex of feasible region

✕Using a solid line for a strict inequality (or vice versa)

✕Shading the wrong side of a boundary

✕Picking a non-integer optimum when the question requires integers

✕Forgetting x≥0, y≥0x \geq 0,\ y \geq 0x≥0, y≥0

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Linear Programming — frequently asked questions

✕Forgetting $x \geq 0,\ y \geq 0$