Class 12 Linear Programming · Very Long Answer5 Marksgraphical_optimization

Class 12 Maths · Linear Programming · Very Long Answer #64

Maximize Z=x + y subject to x + 4*y ≤ 8; 2*x + 3*y ≤ 12; 3*x + y ≤ 9; -x ≤ 0; -y ≤ 0. Solve graphically.

✅ Step-by-Step Solution

Final Answer: max Z=43/11 at (28/11,15/11).

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 + 4*y = 8; 2*x + 3*y = 12; 3*x + y = 9; -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), (0,2), (28/11,15/11), (3,0).
Evaluate the objective at every vertex: (0,0) → 0; (0,2) → 2; (28/11,15/11) → 43/11; (3,0) → 3.
The region is bounded. max Z=43/11 at (28/11,15/11).

[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 + 4*y = 8; 2*x + 3*y = 12; 3*x + y = 9; -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), (0,2), (28/11,15/11), (3,0).
[1 mark] Evaluate the objective at every vertex: (0,0) → 0; (0,2) → 2; (28/11,15/11) → 43/11; (3,0) → 3.
[1 mark] The region is bounded. max Z=43/11 at (28/11,15/11).

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