The minimum value of \(Z = 3x - y\) subject to the constraints \(x + y \ge 5\), \(x \le 4\), \(y \le 6\), \(x \ge 0\), \(y \ge 0\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(-6\)
The region lies above \(x + y = 5\), left of \(x = 4\) and below \(y = 6\), in the first quadrant. It is bounded, with corner points:
(0, 5) and (4, 1) on the line \(x + y = 5\)
(4, 6) and (0, 6) on the line \(y = 6\)
Values of \(Z = 3x - y\):
\(Z(0, 5) = -5\)
\(Z(4, 1) = 11\)
\(Z(4, 6) = 6\)
\(Z(0, 6) = -6\)
The minimum value is \(-6\), at (0, 6). Because y has a negative coefficient, Z is smallest where x is least and y is greatest.
Why the other options are incorrect
Option (2) \(-5\) — Incorrect. This is Z at (0, 5), but (0, 6) gives the smaller value \(-6\).
Option (3) \(6\) — Incorrect. This is Z at (4, 6), which is not the smallest corner value.
Option (4) \(11\) — Incorrect. This is Z at (4, 1), the maximum, not the minimum.