Maximize $LaTeX: \displaystyle p = 8 x + 14 y$ subject to $LaTeX: \displaystyle \begin{cases}97 x + 19 y \leq 1843 \\ 68 x + 85 y \leq 5780 \\ x \geq 0, y \geq 0 \end{cases}$

Drawing a graph gives
Solving the system of equations gives the intersection at $LaTeX: \displaystyle \left( \frac{2755}{409}, \ \frac{25608}{409}\right)$ . Making a table gives to test the verticies in $LaTeX: \displaystyle p=8 x + 14 y$ gives

\begin{tabular}{|c|c|}\hline Point & Function \\[3pt] \hline $LaTeX: \displaystyle \left( 0, \ 0\right)$ & $LaTeX: \displaystyle 0$ \\[3pt] \hline $LaTeX: \displaystyle \left( 19, \ 0\right)$ & $LaTeX: \displaystyle 152$ \\[3pt] \hline $LaTeX: \displaystyle \left( \frac{2755}{409}, \ \frac{25608}{409}\right)$ & $LaTeX: \displaystyle \frac{380552}{409}$ \\[3pt] \hline $LaTeX: \displaystyle \left( 0, \ 68\right)$ & $LaTeX: \displaystyle 952$ \\[3pt] \hline \end{tabular} The gives the maximum value of $LaTeX: \displaystyle 952$ located at $LaTeX: \displaystyle \left( 0, \ 68\right)$ .