Applied Maths. Section B2Linear Programming

The feasible region of the LPP with the objective function z=3x+2y ,…

The feasible region of the LPP with the objective function $ z=3x+2y $, is given as shaded portion in the figure given below:

$Figure of the feasible region$

Which of the following are the constraints for the above LPP ?:

(A) $ y-x+5\ge 0 $

(B) $ y-x+5\le 0 $

(D) $ y\le 2 $

Choose the correct answer from the options given below:

Answer & explanation

Correct answer: option 4

The correct answer is Option (4) → (B) and (C) only

The figure has two boundary lines.

The slanting line passes through (5, 0) and (0, −5), so its equation is

$ \displaystyle \frac{x}{5}+\frac{y}{-5}=1 $

$ \displaystyle x-y=5 $

$ \displaystyle y-x+5=0 $

The horizontal line passes through (0, 2), so its equation is y = 2. The two lines meet at (7, 2).

The shaded region lies above the line y = 2 and below the slanting line. A point of the shaded region, for example (10, 3), gives

$ \displaystyle y-x+5=3-10+5=-2\le 0 $
$ \displaystyle y=3\ge 2 $

So the shaded region satisfies $ y-x+5\le 0 $ and $ y\ge 2 $.

Checking the statements

(A) $ y-x+5\ge 0 $ — Incorrect. This is the side containing the origin (0 − 0 + 5 = 5 ≥ 0), which is not shaded.

(B) $ y-x+5\le 0 $ — Correct. The shaded point (10, 3) gives −2 ≤ 0.

(C) $ y\ge 2 $ — Correct. The shaded region is above the line y = 2.

(D) $ y\le 2 $ — Incorrect. That is the part below the line y = 2, which is not shaded.