The shaded region in the following figure represents the solution set for a certain linear programming…

The shaded region in the following figure represents the solution set for a certain linear programming problem. Then linear constraints for this region are given by
  1. $\begin{aligned} 2 x+3 y \geq 6,-x+2 y & \geq 2,3 x+6 y \leq 18, \\ x-3 y & \geq 3, x \geq 0, y \geq 0\end{aligned}$
  2. $\begin{aligned} 2 x+3 y \geq 6,-x+2 y & \leq 2, x-3 y \leq 3 \\ x+2 y & \geq 18, x \geq 0, y \geq 0\end{aligned}$
  3. $\begin{aligned} 2 x+3 y \leq 6,-x+2 y & \geq 2,3 x+6 y \leq 18 \\ x-3 y & \leq 3, x \geq 0, y \geq 0\end{aligned}$
  4. $\begin{aligned} 2 x+3 y \geq 6,3 x+6 y & \leq 18, x-3 y \leq 3 \\ -x+2 y & \leq 2, x \geq 0, y \geq 0\end{aligned}$

Solution

Shaded region lies on origin side of $3 x+6 y=18, x-3 y=3,-x+2 y=2$ and on non-origin side of $2 x+3 y=6$. $\begin{aligned} \therefore \quad & 2 x+3 y \geq 6,3 x+6 y \leq 18, x-3 y \leq 3, \\ & -x+2 y \leq 2, x \geq 0, y \geq 0 \end{aligned}$

Asked in: MHT CET 2023 (14 May Shift 2)

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