For which of the following objective functions does Z have the same value at the points (2, 3) and (5, 1)?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(Z = 2x + 3y\)
Z has the same value at two points when both lie on one line \(ax + by = k\), a line of constant Z. For \(Z = 2x + 3y\):
\(Z(2, 3) = 4 + 9 = 13\)
\(Z(5, 1) = 10 + 3 = 13\)
So both points lie on the line \(2x + 3y = 13\).
Why the other options are incorrect
Option (2) \(Z = 3x + 2y\) — Incorrect. Z(2, 3) = 12 and Z(5, 1) = 17.
Option (3) \(Z = x + 3y\) — Incorrect. Z(2, 3) = 11 and Z(5, 1) = 8.
Option (4) \(Z = 3x + y\) — Incorrect. Z(2, 3) = 9 and Z(5, 1) = 16.