Consider the LPP: Maximize \(Z = x + y\) subject to the constraints \(x + 2y \le 70, 2x + y \le 95, x, y \ge 0\). The optimal feasible solution is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (40, 15)
The feasible region is bounded, so the maximum of Z occurs at a corner point.
\(x + 2y = 70\) meets the axes at (70, 0) and (0, 35).
\(2x + y = 95\) meets the axes at (47.5, 0) and (0, 95).
The two lines meet where \(y = 95 - 2x\):
\(x + 2(95 - 2x) = 70\)
\(-3x = -120,\quad x = 40,\quad y = 15\)
The corners of the feasible region are (0, 0), (47.5, 0), (40, 15) and (0, 35).
\(Z(0, 0) = 0\)
\(Z(47.5, 0) = 47.5\)
\(Z(40, 15) = 55\)
\(Z(0, 35) = 35\)
The maximum, 55, is at (40, 15).
Why the other options are incorrect
Every option gives Z = 55, so the deciding test is feasibility: each wrong option breaks the constraint \(x + 2y \le 70\).
Option (1) (20, 35) — Incorrect. \(x + 2y = 20 + 70 = 90 \gt 70\), so the point is not feasible.
Option (2) (35, 20) — Incorrect. \(x + 2y = 35 + 40 = 75 \gt 70\), so the point is not feasible.
Option (3) (30, 25) — Incorrect. \(x + 2y = 30 + 50 = 80 \gt 70\), so the point is not feasible.