Which one of the region I, II, III, IV is represented by the constraints \( x+3y\le 60,x+y\ge 10,x\le y,x\ge 0,y\ge 0 \)
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Region I , Quadrilateral ABEF
Each constraint is checked with a test point on one side of its line.
\( \displaystyle x+3y\le 60:\ (0,0)\text{ satisfies it, so the region is on the origin side of FD.} \)
\( \displaystyle x+y\ge 10:\ (0,0)\text{ does not satisfy it, so the region is on the side of AC away from the origin.} \)
\( \displaystyle x\le y:\ (0,1)\text{ satisfies it, so the region is above the line }y=x\text{ (the line OBE).} \)
With x ≥ 0 and y ≥ 0, the common region lies above OBE, beyond AC and below FD. Its corners are
\( \displaystyle A(0,10),\ B(5,5),\ E(15,15),\ F(0,20) \)
This is Region I, the quadrilateral ABEF.
Why the other options are incorrect
Option (2) Region II, Δ OAB — Incorrect. Region II, ΔOAB, is on the origin side of AC, so it has x + y ≤ 10.
Option (3) Region III, Δ OCB — Incorrect. Region III, ΔOCB, is below the line y = x and on the origin side of AC, so it breaks both x ≤ y and x + y ≥ 10.
Option (4) Region IV, Quadrilateral BCDE — Incorrect. Region IV, quadrilateral BCDE, is below the line y = x, so it has x ≥ y.
Tip: For a line through the origin such as y = x, the origin cannot be used as a test point; use a point like (0, 1) instead.