The minimum value of \(Z = 13x - 15y\) subject to the constraints \(x + y \le 7\), \(2x - 3y + 6 \ge 0\), \(x \ge 0\), \(y \ge 0\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → −30
The line \(2x - 3y + 6 = 0\) meets the y-axis at (0, 2), and the region lies on the side of the origin, below it. It meets \(x + y = 7\) where, putting \(y = 7 - x\),
\(2x - 21 + 3x + 6 = 0\)
\(x = 3,\ y = 4\)
The corner points are (0, 0), (7, 0), (3, 4) and (0, 2). Values of Z:
\(Z(0, 0) = 0\)
\(Z(7, 0) = 91\)
\(Z(3, 4) = 39 - 60 = -21\)
\(Z(0, 2) = -30\)
The region is bounded, so the minimum value of Z is −30, at (0, 2).
Why the other options are incorrect
Option (2) −21 — Incorrect. This is Z at (3, 4), which is more than −30.
Option (3) 0 — Incorrect. This is Z at the origin; the negative coefficient of y makes smaller values possible.
Option (4) 91 — Incorrect. This is the maximum value, at (7, 0).