Class 12 Linear Programming · Very Long Answer5 Marksgraphical_optimization

Class 12 Maths · Linear Programming · Very Long Answer #69

Minimize Z=3*x + 5*y subject to -x - 2*y ≤ -10; -x - y ≤ -6; -3*x - y ≤ -8; -x ≤ 0; -y ≤ 0. Solve graphically.

✅ Step-by-Step Solution

Final Answer: min Z=26 at (2,4).

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 - 2*y = -10; -x - y = -6; -3*x - y = -8; -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: (0,8), (1,5), (2,4), (10,0).
Evaluate the objective at every vertex: (0,8) → 40; (1,5) → 28; (2,4) → 26; (10,0) → 30.
The region is unbounded. Its recession directions satisfy every homogeneous constraint; representatives: (1,0), (0,1), (0,1), (1,0). Their objective changes are 3, 5, 5, 3. This checks whether a finite optimum exists. min Z=26 at (2,4).

[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 - 2*y = -10; -x - y = -6; -3*x - y = -8; -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: (0,8), (1,5), (2,4), (10,0).
[1 mark] Evaluate the objective at every vertex: (0,8) → 40; (1,5) → 28; (2,4) → 26; (10,0) → 30.
[1 mark] The region is unbounded. Its recession directions satisfy every homogeneous constraint; representatives: (1,0), (0,1), (0,1), (1,0). Their objective changes are 3, 5, 5, 3. This checks whether a finite optimum exists. min Z=26 at (2,4).

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