Class 12 Linear Programming · Long Answer3 Marksgraphical_optimization

Class 12 Maths · Linear Programming · Long Answer #28

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

✅ Step-by-Step Solution

Final Answer: min Z=3 at (0,3).

Draw each boundary ax+by=c and retain the closed half-plane ax+by≤c; intersect all constraints, including x,y≥0.
Solving pairs of boundary equations and rejecting points violating any inequality gives vertices: (0,3), (2,1), (4,0).
Evaluate the objective at every vertex: (0,3) → 3; (2,1) → 9; (4,0) → 16.
The region is unbounded. Its recession directions satisfy every homogeneous constraint; representatives: (1,0), (0,1), (0,1), (1,0). Their objective changes are 4, 1, 1, 4. This checks whether a finite optimum exists. min Z=3 at (0,3).

[0.5 marks] Draw each boundary ax+by=c and retain the closed half-plane ax+by≤c; intersect all constraints, including x,y≥0.
[1 mark] Solving pairs of boundary equations and rejecting points violating any inequality gives vertices: (0,3), (2,1), (4,0).
[1 mark] Evaluate the objective at every vertex: (0,3) → 3; (2,1) → 9; (4,0) → 16.
[0.5 marks] The region is unbounded. Its recession directions satisfy every homogeneous constraint; representatives: (1,0), (0,1), (0,1), (1,0). Their objective changes are 4, 1, 1, 4. This checks whether a finite optimum exists. min Z=3 at (0,3).

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