Class 12 Linear Programming · Very Long Answer5 Marksgraphical_optimization

Class 12 Maths · Linear Programming · Very Long Answer #62

Minimize Z=x - 5*y + 20 subject to -x + y ≤ 0; x - 2*y ≤ -2; -x ≤ -3; y ≤ 4; -x ≤ 0; -y ≤ 0. Solve graphically.

✅ Step-by-Step Solution

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

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 + y = 0; x - 2*y = -2; -x = -3; y = 4; -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: (3,5/2), (3,3), (4,4), (6,4).
Evaluate the objective at every vertex: (3,5/2) → 21/2; (3,3) → 8; (4,4) → 4; (6,4) → 6.
The region is bounded. min Z=4 at (4,4).

[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 + y = 0; x - 2*y = -2; -x = -3; y = 4; -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: (3,5/2), (3,3), (4,4), (6,4).
[1 mark] Evaluate the objective at every vertex: (3,5/2) → 21/2; (3,3) → 8; (4,4) → 4; (6,4) → 6.
[1 mark] The region is bounded. min Z=4 at (4,4).

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