subject to the constraints: 
.Solution:
It is given that: 
and the constraints 
, 
Taking 
, we have
![\[\begin{tabular}{|l|l|l|} \hline$x$ & 4 & 3 \\ \hline $\mathrm{v}$ & 3 & 4 \\ \hline \end{tabular}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a89c1e28a02ef115f82e3f5e78e95c4b_l3.png "Rendered by QuickLaTeX.com")
And, taking 
we have
![\[\begin{tabular}{|l|l|l|} \hline$x$ & 1 & $-3$ \\ \hline$y$ & 2 & 0 \\ \hline \end{tabular}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-fc615c7ae759639a0eb48710155bac4b_l3.png "Rendered by QuickLaTeX.com")
Now, plotting all the constrain equations it can be seen that the shaded area 
is the feasible region determined by the constraints.
The feasible reqion is bounded with four corners 
and 
Therefore, the maximum value can occur at any corner.
On evaluating the value of 
, we obtain
![\[\begin{tabular}{|l|l|} \hline Corner points & Value of $\mathrm{Z}$ \\ \hline $\mathrm{O}(0,0)$ & $13(0)-15(0)=0$ \\ \hline $\mathrm{A}(7,0)$ & $13(7)-15(0)=91$ \\ \hline $\mathrm{B}(3,4)$ & $13(3)-15(4)=-21$ \\ \hline $\mathrm{C}(0,2)$ & $13(0)-15(2)=-30$ \\ \hline \end{tabular}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-65eeddd2f7496352bd37c796573f6a2c_l3.png "Rendered by QuickLaTeX.com")
It can be seen from the above table it’s seen that the minimum value of is 
. Therefore, the minimum value of the function is at 
.