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The objective function $\mathrm{z}=4 \mathrm{z}+5 \mathrm{y}$ subjective to $2 \mathrm{x}+\mathrm{y} \geq 7$; $2 x+3 y \leq 15 ; y \leq 3, x \geq 0 ; y \geq 0$ has minimum value at the point.
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Verified Answer
The correct answer is:
on X-axis
We have lines $2 x+y=7,2 x+3 y=15, y=3$ Refer figure
The required region is shaded.
We have $\mathrm{A} \equiv\left(\frac{7}{2}, 0\right), \mathrm{B} \equiv\left(\frac{15}{2}, 0\right)$
Point of intersection of $2 x+y=7$ and $y=3$ is $D=(2,3)$
$$
\begin{aligned}
& \mathrm{z}_{(\mathrm{A})}=4\left(\frac{7}{2}\right)+5(0)=14+0=14 \\
& \mathrm{z}_{(\mathrm{B})}=5\left(\frac{15}{2}\right)+5(0)=30+0=30 \\
& \mathrm{z}_{(\mathrm{C})}=4(3)+5(3)=12+15=27 \\
& \mathrm{z}_{(\mathrm{D})}=4(2)+5(3)=8+15=23
\end{aligned}
$$
Hence minimum value occurs at point a which lies on $\mathrm{X}$ axis.
The required region is shaded.
We have $\mathrm{A} \equiv\left(\frac{7}{2}, 0\right), \mathrm{B} \equiv\left(\frac{15}{2}, 0\right)$
Point of intersection of $2 x+y=7$ and $y=3$ is $D=(2,3)$
$$
\begin{aligned}
& \mathrm{z}_{(\mathrm{A})}=4\left(\frac{7}{2}\right)+5(0)=14+0=14 \\
& \mathrm{z}_{(\mathrm{B})}=5\left(\frac{15}{2}\right)+5(0)=30+0=30 \\
& \mathrm{z}_{(\mathrm{C})}=4(3)+5(3)=12+15=27 \\
& \mathrm{z}_{(\mathrm{D})}=4(2)+5(3)=8+15=23
\end{aligned}
$$
Hence minimum value occurs at point a which lies on $\mathrm{X}$ axis.
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