The objective function of a LPP is given by \( z=ax+by \), where \( a \) and \( b \) are constants. If minimum of \( z \) occurs at two points (50, 30) and (20, 40) , then
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( 3a-b=0 \)
If the minimum occurs at both points, z has the same value at them.
\( \displaystyle z(50,30)=z(20,40) \)
\( \displaystyle 50a+30b=20a+40b \)
\( \displaystyle 30a=10b \)
\( \displaystyle 3a=b \)
\( \displaystyle 3a-b=0 \)
Why the other options are incorrect
Option (1) \( 3b=a \) — Incorrect. This interchanges a and b; from 30a = 10b the relation is b = 3a.
Option (2) \( 3b+a=0 \) — Incorrect. This gives a = −3b, but the relation is b = 3a.
Option (3) \( 3a+b=0 \) — Incorrect. The sign is wrong: 30a − 10b = 0 gives 3a − b = 0.