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Find the maximum and minimum values of $Z=2 x+y$, subject to the constraints $x+3 y \geq 6, x-3 y \leq 3,3 x+4 y \leq 24$ $-3 x+2 y \leq 6,5 x+y \geq 5, x \geq 0$ and $y \geq 0$
Find the maximum and minimum values of $Z=2 x+y$, subject to the constraints $x+3 y \geq 6, x-3 y \leq 3,3 x+4 y \leq 24$ $-3 x+2 y \leq 6,5 x+y \geq 5, x \geq 0$ and $y \geq 0$
CBSE | Class 12 | Exercise 33B | Linear Programming | Maths | RS Aggarwal
Find the maximum and minimum values of $Z=2 x+y$, subject to the constraints $x+3 y \geq 6, x-3 y \leq 3,3 x+4 y \leq 24$ $-3 x+2 y \leq 6,5 x+y \geq 5, x \geq 0$ and $y \geq 0$

The feasible region determined by $x+3 y \geq 6, x-3 y \leq 3,3 x+4 y \leq 24$
$-3 x+2 y \leq 6,5 x+y \geq 5, x \geq 0$ and $y \geq 0$ is given by

The corner points of the feasible region are $A(4 / 3,5), B(4 / 13,45 / 13), C(9 / 14,25 / 14), D(9 / 2,1 / 2), E(84 / 13,15 / 13)$. The value of $Z$ at corner points are
$$

\begin{tabular}{|l|l|l|}
\hline Corner Point & $\mathrm{Z}=2 \mathrm{x}+\mathrm{y}$ & \\
\hline $\mathrm{A}(4 / 3,5)$ & $23 / 3$ & \\
\hline $\mathrm{B}(4 / 13,45 / 13)$ & $53 / 13$ & \\
\hline $\mathrm{C}(9 / 14,25 / 14)$ & $43 / 14$ & Minimum \\
\hline $\mathrm{D}(9 / 2,1 / 2)$ & $19 / 2$ & \\
\hline $\mathrm{E}(84 / 13,15 / 13)$ & $183 / 13$ & Maximum \\
\hline
\end{tabular}

$$

The maximum and minimum value of $Z$ is $183 / 13$ and $43 / 14$ at points $E(84 / 13,15 / 13)$ and $C(9 / 14,25 / 14)$.

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