Class 12 Linear Programming · Very Long Answer5 Marksgraphical_optimization

Class 12 Maths · Linear Programming · Very Long Answer #43

Maximize Z=15*x + 10*y subject to 2*x + y ≤ 40; 2*x + 3*y ≤ 80; -x ≤ 0; -y ≤ 0. Solve graphically.

✅ Step-by-Step Solution

Final Answer: max Z=350 at (10,20).

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 + y = 40; 2*x + 3*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), (0,80/3), (10,20), (20,0).
Evaluate the objective at every vertex: (0,0) → 0; (0,80/3) → 800/3; (10,20) → 350; (20,0) → 300.
The region is bounded. max Z=350 at (10,20).

[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 + y = 40; 2*x + 3*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), (0,80/3), (10,20), (20,0).
[1 mark] Evaluate the objective at every vertex: (0,0) → 0; (0,80/3) → 800/3; (10,20) → 350; (20,0) → 300.
[1 mark] The region is bounded. max Z=350 at (10,20).

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