Maximize $LaTeX: \displaystyle p = 4 x + 3 y$ subject to $LaTeX: \displaystyle \begin{cases}97 x + 7 y \leq 679 \\ 85 x + 40 y \leq 3400 \\ x \geq 0, y \geq 0 \end{cases}$

Drawing a graph gives
Solving the system of equations gives the intersection at $LaTeX: \displaystyle \left( \frac{224}{219}, \ \frac{18139}{219}\right)$ . Making a table gives to test the verticies in $LaTeX: \displaystyle p=4 x + 3 y$ gives

\begin{tabular}{|c|c|}\hline Point & Function \\[3pt] \hline $LaTeX: \displaystyle \left( 0, \ 0\right)$ & $LaTeX: \displaystyle 0$ \\[3pt] \hline $LaTeX: \displaystyle \left( 7, \ 0\right)$ & $LaTeX: \displaystyle 28$ \\[3pt] \hline $LaTeX: \displaystyle \left( \frac{224}{219}, \ \frac{18139}{219}\right)$ & $LaTeX: \displaystyle \frac{55313}{219}$ \\[3pt] \hline $LaTeX: \displaystyle \left( 0, \ 85\right)$ & $LaTeX: \displaystyle 255$ \\[3pt] \hline \end{tabular} The gives the maximum value of $LaTeX: \displaystyle 255$ located at $LaTeX: \displaystyle \left( 0, \ 85\right)$ .