Class 12 Linear Programming · Very Long Answer5 Marksgraphical_optimization

Class 12 Maths · Linear Programming · Very Long Answer #65

Maximize Z=1000*x + 600*y subject to x + y ≤ 200; -x ≤ -20; 4*x - y ≤ 0; -x ≤ 0; -y ≤ 0. Solve graphically.

✅ Step-by-Step Solution

Final Answer: max Z=136000 at (40,160).

Draw each boundary ax+by=c and retain the closed half-plane ax+by≤c; intersect all constraints, including x,y≥0.
Boundary equations: x + y = 200; -x = -20; 4*x - y = 0; -x = 0; -y = 0. Use an off-line test point to choose each half-plane.
Solving pairs of boundary equations and rejecting points violating any inequality gives vertices: (20,80), (20,180), (40,160).
Evaluate the objective at every vertex: (20,80) → 68000; (20,180) → 128000; (40,160) → 136000.
The region is bounded. max Z=136000 at (40,160).

[1 mark] Draw each boundary ax+by=c and retain the closed half-plane ax+by≤c; intersect all constraints, including x,y≥0.
[1 mark] Boundary equations: x + y = 200; -x = -20; 4*x - y = 0; -x = 0; -y = 0. Use an off-line test point to choose each half-plane.
[1 mark] Solving pairs of boundary equations and rejecting points violating any inequality gives vertices: (20,80), (20,180), (40,160).
[1 mark] Evaluate the objective at every vertex: (20,80) → 68000; (20,180) → 128000; (40,160) → 136000.
[1 mark] The region is bounded. max Z=136000 at (40,160).

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