Class 12 Linear Programming · Very Long Answer5 Marksgraphical_optimization

Class 12 Maths · Linear Programming · Very Long Answer #70

Maximize Z=x + y subject to 5*x ≤ 4; 3*x + y ≤ 5; x + 2*y ≤ 6; -x ≤ 0; -y ≤ 0. Solve graphically.

✅ Step-by-Step Solution

Final Answer: max Z=17/5 at (4/5,13/5).

Draw each boundary ax+by=c and retain the closed half-plane ax+by≤c; intersect all constraints, including x,y≥0.
Boundary equations: 5*x = 4; 3*x + y = 5; x + 2*y = 6; -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), (0,3), (4/5,0), (4/5,13/5).
Evaluate the objective at every vertex: (0,0) → 0; (0,3) → 3; (4/5,0) → 4/5; (4/5,13/5) → 17/5.
The region is bounded. max Z=17/5 at (4/5,13/5).

[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: 5*x = 4; 3*x + y = 5; x + 2*y = 6; -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), (0,3), (4/5,0), (4/5,13/5).
[1 mark] Evaluate the objective at every vertex: (0,0) → 0; (0,3) → 3; (4/5,0) → 4/5; (4/5,13/5) → 17/5.
[1 mark] The region is bounded. max Z=17/5 at (4/5,13/5).

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