Class 12 Mathematics Chapter 12: Linear Programming

CBSE 2026–27 | Detailed NCERT-aligned free resource

Linear Programming converts a practical optimisation problem into inequalities, a feasible region and an objective function. The graphical method for two variables is the central skill.

1. Key Terms

• Decision variables: quantities to be determined.
• Objective function: expression to maximise or minimise.
• Constraints: restrictions represented by linear inequalities.
• Feasible region: common region satisfying all constraints.
• Optimal solution: feasible point giving the required maximum or minimum.

2. Translating a Word Problem

Suppose x represents units of product A and y represents units of product B. If total material cannot exceed 40 units, write an inequality such as ax+by≤40. Repeat for every resource restriction.

3. Graphing an Inequality

First draw the boundary line by replacing ≤ or ≥ with =. Test a point, usually (0,0), to decide which side is feasible.

4. Corner-Point Method

For a linear programming problem with a bounded feasible region, evaluate the objective function at the corner points. The required optimum occurs at a corner point under the standard graphical framework.

Worked Example

Maximise Z=3x+2y subject to x+y≤4, x≤2, y≤3, x≥0,y≥0. The feasible vertices are (0,0), (2,0), (2,2), (1,3), (0,3). Evaluate Z: 0,6,10,9,6. Hence the maximum value within this feasible region is 10 at (2,2).

5. Bounded and Unbounded Regions

A feasible region may be bounded or unbounded. An unbounded region does not automatically mean that an optimum does not exist; the objective function and direction must be examined.

6. Redundant Constraints

A constraint is redundant when removing it does not change the feasible region. Identifying redundancy can simplify the graphical solution.

7. Practical Checklist

1. Define x and y with units.
2. Write the objective function.
3. Translate every condition into an inequality.
4. Include non-negativity restrictions.
5. Draw boundary lines and shade the common feasible region.
6. Find all corner points.
7. Evaluate Z at every relevant corner point.
8. State the optimum with units and interpretation.

8. Common Mistakes

• Shading the wrong side of an inequality.
• Forgetting x≥0 and y≥0.
• Missing a corner point.
• Giving only the maximum/minimum value without identifying the corresponding quantities.

9. Practice Set

1. Convert a production problem into an LPP.
2. Graph three linear inequalities and identify the feasible region.
3. Maximise a profit function at the corner points.
4. Solve a minimisation problem graphically.
5. Identify a redundant constraint in a given system.