Q1. Maximise Z = 3x + 4y subject to x + y ≤ 4, x ≥ 0, y ≥ 0.
Corners of the feasible region: (0,0), (4,0), (0,4)
Z at (0,0) = 0; Z at (4,0) = 12; Z at (0,4) = 16
Maximum Z = 16 at (0, 4).
Linear Programming — Class 12
Linear programming finds the best outcome (maximum profit, minimum cost) within a set of straight-line constraints — using just the corners of a region.
Objective functionThe linear expression to maximise or minimise, e.g. Z = 3x + 5y (profit or cost).
ConstraintsLinear inequalities that restrict x and y (e.g. resource limits); together they define the feasible region.
Feasible regionThe set of all (x, y) satisfying every constraint — always a convex polygon (for bounded problems).
Corner point methodThe optimum value of Z always occurs at a CORNER (vertex) of the feasible region — evaluate Z at each corner and compare.
Solved examples
Formula sheet
| Quantity | Formula |
|---|---|
| Objective function | Z = ax + by |
| Corner point theorem | optimum Z occurs at a vertex of the feasible region |
Common mistakes & exam wins
- The optimum (max or min) value ALWAYS occurs at a corner point of the feasible region — never inside it.
- Plot each constraint as a line, shade the region satisfying all of them, then find where the boundary lines intersect.
- Evaluate the objective function at EVERY corner point and simply compare the values.
Frequently asked questions
What is a linear programming problem?
A problem of maximising or minimising a linear objective function subject to a set of linear constraints (inequalities).
What is the corner point method?
A method that finds the optimal value of the objective function by checking its value only at the corners (vertices) of the feasible region.
What is the feasible region?
The set of all points (x, y) that satisfy every constraint of the problem simultaneously.