What it shows
A linear programming problem asks for the largest (or smallest) value of a linear objective Z = px + qy, when x and y must obey a few linear constraints such as 2x + y ≤ 20, and x, y ≥ 0. Every point that obeys all the constraints together makes the feasible region, a polygon. The corner point theorem says that if Z has a best value on the region, it is reached at a corner. So you draw the region, list its corners, and work out Z at each one.
This page does all of that live: it shades the region, labels the corners A, B, C … in a table with x, y and Z, highlights the best one, and lets you slide the dashed objective line Z = k across the region to see why the best point is a corner. It also spots an unbounded region, a problem with no maximum, ties along an edge, and constraints that cannot all be met.
It follows linear programming in Class 12 maths (the graphical method), and builds on drawing straight lines and solving pairs of equations, which the linear pairs covers. For more than two variables, whole-number answers, shadow prices and the simplex method, go on to the LP solver; the transportation solver tool solves a classic special case.
How to use it
- Pick a story, or My own. The stories are making tables and chairs, a cheapest diet, planting two crops, a tie, and a problem with no feasible point.
- Choose Maximise or Minimise and type the numbers in Z = p x + q y.
- Edit the constraints: the x and y coefficients, ≤ or ≥, and the right side. Add a constraint (up to five) or remove one with ✕. x ≥ 0 and y ≥ 0 always apply.
- Read the picture: each constraint line is numbered, the shaded area is the feasible region, and green dots mark the best corner.
- Move the objective line slider, or press Slide to the best, to watch Z = k leave the region at the optimum.
- The table and the sentence under it give the answer.
Worked example
The tables and chairs story: maximise Z = 50x + 30y with 2x + y ≤ 20 (machine hours), x + 2y ≤ 16 (finishing hours), x, y ≥ 0. The corners are (0, 0), (10, 0), (8, 4) and (0, 8), with Z = 0, 500, 520 and 240. The page shows Maximum Z = 520 at x = 8 tables and y = 4 chairs.
The diet story: minimise Z = 4x + 6y with 2x + y ≥ 8 and x + 3y ≥ 9. The region is unbounded, with corners (0, 8), (3, 2) and (9, 0) giving Z = 48, 24 and 36. The page shows Minimum Z = 24 at (3, 2) and explains that the half-plane 4x + 6y < 24 has no point in the region, so 24 really is the minimum. Switch to Maximise and the page says No maximum.
How it works
The region is built by starting with a huge square in the first quadrant and cutting it down with each constraint’s half-plane. Corners that land on the far edges of that square show the region goes on for ever. If the best Z is at one of those far corners, Z keeps improving without limit, so there is no maximum (or minimum). Otherwise the best real corner is the answer, and two best corners side by side mean every point on the edge between them is optimal.
Check yourself
Five fresh questions: where a line cuts an axis, Z at a corner, a full small problem, the unbounded-region rule and whether a point is feasible. Nothing is saved.
FAQ
What about equality constraints? Write x + y = 5 as two constraints, x + y ≤ 5 and x + y ≥ 5; the region becomes a segment.
Why must x and y be at least 0? In real problems they count things such as tables or kilograms, which cannot be negative.
Is this from my textbook? It covers the standard school topic with its own examples. It is an independent study aid and is not part of NCERT.
Where the numbers come from
- Standard school mathematics: linear programming problems in two variables solved by the graphical (corner point) method, including the half-plane test for unbounded regions. Written for fibub; no textbook text, figures or exercises are used, and the story numbers are made up.
- Checked against scipy.optimize.linprog (HiGHS) for seven problems: bounded maximum, unbounded region with a minimum, unbounded maximum, a tie along an edge, an infeasible pair and a three-constraint problem minimised and maximised.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.