knowledge-base/class_12/ch12/0108_algebra.md

metadata

source: ncert
topic: algebra
class_level: class_12
chapter: ch12
difficulty: advanced

Class 12 | Ch12: Linear Programming LINEAR PROGRAMMING PROBLEM (LPP): Objective function: Z=ax+by (to maximise or minimise). Constraints: inequalities (ax+by<=c, x>=0, y>=0 etc). Feasible region: set of all points satisfying ALL constraints simultaneously. Feasible solution: any point in feasible region. Optimal solution: feasible solution that gives maximum/minimum value of Z. CORNER POINT METHOD (THEOREM): If optimal solution exists, it occurs at a CORNER POINT (vertex) of feasible region. Steps:

1. Graph all constraints. Find feasible region.
2. Find all corner points (vertices) of feasible region.
3. Evaluate Z at each corner point.
4. Maximum/minimum value of Z is the optimal solution. TYPES OF LPP: Bounded feasible region: has maximum AND minimum. Unbounded feasible region: may not have maximum (for maximise) or minimum (for minimise). Infeasible: no feasible region (constraints contradictory). No solution. FORMULATING LPP FROM WORD PROBLEMS: Identify: variables (what to find), objective (what to optimise), constraints (conditions). SOLVED EXAMPLES: Example 1: Maximise Z=3x+4y subject to x+y<=450, 2x+y<=600, x>=0, y>=0. Corner points: O(0,0), A(300,0), B(150,300), C(0,450). Z at O=0. Z at A=900. Z at B=450+1200=1650. Z at C=1800. Maximum Z=1800 at C(0,450). Example 2: Minimise Z=5x+7y subject to 2x+y>=8, x+2y>=10, x>=0, y>=0. Corner points: A(0,8), B(2,4), C(10,0). Z at A=56. Z at B=10+28=38. Z at C=50. Minimum Z=38 at B(2,4). FINDING CORNER POINTS: solve pairs of boundary equations simultaneously. Intersection of 2x+y=600 and x+y=450: subtract → x=150, y=300. COMMON MISTAKES: Evaluate Z at ALL corner points (not just one or two). Feasible region must satisfy ALL constraints (including x>=0, y>=0). For minimum in unbounded region: check if solution is truly minimum.