Class 12 Linear Programming · Long Answer3 Marksgraphical_optimization

Class 12 Maths · Linear Programming · Long Answer #32

Minimize Z=5*x + 10*y subject to x + 2*y ≤ 120; -x - y ≤ -60; -x + 2*y ≤ 0; -x ≤ 0; -y ≤ 0. Solve graphically.

✅ Step-by-Step Solution

Final Answer: min Z=300 at (60,0).

Draw each boundary ax+by=c and retain the closed half-plane ax+by≤c; intersect all constraints, including x,y≥0.
Solving pairs of boundary equations and rejecting points violating any inequality gives vertices: (40,20), (60,0), (60,30), (120,0).
Evaluate the objective at every vertex: (40,20) → 400; (60,0) → 300; (60,30) → 600; (120,0) → 600.
The region is bounded. min Z=300 at (60,0).

[0.5 marks] Draw each boundary ax+by=c and retain the closed half-plane ax+by≤c; intersect all constraints, including x,y≥0.
[1 mark] Solving pairs of boundary equations and rejecting points violating any inequality gives vertices: (40,20), (60,0), (60,30), (120,0).
[1 mark] Evaluate the objective at every vertex: (40,20) → 400; (60,0) → 300; (60,30) → 600; (120,0) → 600.
[0.5 marks] The region is bounded. min Z=300 at (60,0).

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