Integer solutions in regions

Objective. Find and optimise integer solutions in feasible regions.

1. Learn
2. Worked example
3. Practice
4. Feedback
5. Continue

Learn: the key idea

Integer solutions in regions extend feasible-region work by requiring whole-number coordinate points, often to optimise a quantity like cost or profit subject to constraints. After shading the feasible region defined by the inequalities, list the lattice points (integer x and y pairs) that lie within or on its boundary, then evaluate the objective expression at each to find the optimum. For instance, within x + y ≤ 5, x ≥ 1, y ≥ 1 the integer point maximising x + 2y is found by checking candidates like (1,4).

Worked example

Find the integer solution maximising P = x + 3y subject to x ≥ 0, y ≥ 0 and x + y ≤ 4.

1 $List integer points with x + y \leq 4, x \geq 0, y \geq 0$
2 $To maximise P = x + 3y, favour larger y since it has the bigger coefficient$
3 $Largest y with x \geq 0 is y = 4 when x = 0, giving P = 0 + 12 = 12$
4 $Check nearby points like (1,3): P = 1 + 9 = 10, which is smaller, confirming (0,4) is best$

Answer: x = 0, y = 4, giving maximum P = 12

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. Students sometimes find the optimum of the continuous region using a vertex and assume it is already an integer solution without checking it lies on whole-number coordinates. Always test nearby lattice points inside the region if the vertex is non-integer.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard