The objective function is Z=ax+by, and the minimum value occurs at two points: (3,5) and (5,3).
When the minimum value occurs at two different points, the objective function must have the same value at both points.
At point (3,5):
Z=a(3)+b(5)
Z=3a+5b
At point (5,3):
Z=a(5)+b(3)
Z=5a+3b
Since both points give the minimum value:
3a+5b=5a+3b
3a−5a=3b−5b
−2a=−2b
a=b
Therefore, a=b.