Class 12 Linear Programming · Long Answer3 Marksgraphical_optimization

Class 12 Maths · Linear Programming · Long Answer #31

Maximize Z=3*x + 9*y subject to -x - y ≤ -10; x + 3*y ≤ 60; x - y ≤ 0; -x ≤ 0; -y ≤ 0. Solve graphically.

✅ Step-by-Step Solution

Final Answer: max Z=180 at (0,20), (15,15) and every point on their joining segment.

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: (0,10), (0,20), (5,5), (15,15).
Evaluate the objective at every vertex: (0,10) → 90; (0,20) → 180; (5,5) → 60; (15,15) → 180.
The region is bounded. max Z=180 at (0,20), (15,15) and every point on their joining segment.

[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: (0,10), (0,20), (5,5), (15,15).
[1 mark] Evaluate the objective at every vertex: (0,10) → 90; (0,20) → 180; (5,5) → 60; (15,15) → 180.
[0.5 marks] The region is bounded. max Z=180 at (0,20), (15,15) and every point on their joining segment.

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