Class 12 Linear Programming · Very Long Answer5 Marksgraphical_optimization

Class 12 Maths · Linear Programming · Very Long Answer #66

Maximize Z=4*x + y subject to x + y ≤ 50; 3*x + y ≤ 90; -x ≤ -10; -x ≤ 0; -y ≤ 0. Solve graphically.

✅ Step-by-Step Solution

Final Answer: max Z=120 at (30,0).

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 = 50; 3*x + y = 90; -x = -10; -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: (10,0), (10,40), (20,30), (30,0).
Evaluate the objective at every vertex: (10,0) → 40; (10,40) → 80; (20,30) → 110; (30,0) → 120.
The region is bounded. max Z=120 at (30,0).

[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 = 50; 3*x + y = 90; -x = -10; -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: (10,0), (10,40), (20,30), (30,0).
[1 mark] Evaluate the objective at every vertex: (10,0) → 40; (10,40) → 80; (20,30) → 110; (30,0) → 120.
[1 mark] The region is bounded. max Z=120 at (30,0).

Practise interactive questions, flashcards, visual mind maps, and timed board tests for Linear Programming.