Class 12 Linear Programming · Very Long Answer5 Marksgraphical_optimization

Class 12 Maths · Linear Programming · Very Long Answer #72

Maximize Z=7*x/10 + y subject to 2*x + 3*y ≤ 120; 2*x + y ≤ 80; -x ≤ 0; -y ≤ 0. Solve graphically.

✅ Step-by-Step Solution

Final Answer: max Z=41 at (30,20).

Draw each boundary ax+by=c and retain the closed half-plane ax+by≤c; intersect all constraints, including x,y≥0.
Boundary equations: 2*x + 3*y = 120; 2*x + y = 80; -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: (0,0), (0,40), (30,20), (40,0).
Evaluate the objective at every vertex: (0,0) → 0; (0,40) → 40; (30,20) → 41; (40,0) → 28.
The region is bounded. max Z=41 at (30,20).

[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: 2*x + 3*y = 120; 2*x + y = 80; -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: (0,0), (0,40), (30,20), (40,0).
[1 mark] Evaluate the objective at every vertex: (0,0) → 0; (0,40) → 40; (30,20) → 41; (40,0) → 28.
[1 mark] The region is bounded. max Z=41 at (30,20).

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