Class 12 Linear Programming · Very Long Answer5 Marksgraphical_optimization

Class 12 Maths · Linear Programming · Very Long Answer #46

Maximize Z=8*x + 9*y subject to 2*x + 3*y ≤ 6; 3*x - 2*y ≤ 6; y ≤ 1; -x ≤ 0; -y ≤ 0. Solve graphically.

✅ Step-by-Step Solution

Final Answer: max Z=294/13 at (30/13,6/13).

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 = 6; 3*x - 2*y = 6; y = 1; -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), (0,1), (3/2,1), (2,0), (30/13,6/13).
Evaluate the objective at every vertex: (0,0) → 0; (0,1) → 9; (3/2,1) → 21; (2,0) → 16; (30/13,6/13) → 294/13.
The region is bounded. max Z=294/13 at (30/13,6/13).

[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 = 6; 3*x - 2*y = 6; y = 1; -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), (0,1), (3/2,1), (2,0), (30/13,6/13).
[1 mark] Evaluate the objective at every vertex: (0,0) → 0; (0,1) → 9; (3/2,1) → 21; (2,0) → 16; (30/13,6/13) → 294/13.
[1 mark] The region is bounded. max Z=294/13 at (30/13,6/13).

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