Maximize $LaTeX: \displaystyle p = 17 x + 19 y$ subject to $LaTeX: \displaystyle \begin{cases}85 x + 11 y \leq 935 \\ 53 x + 19 y \leq 1007 \\ x \geq 0, y \geq 0 \end{cases}$

Drawing a graph gives
Solving the system of equations gives the intersection at $LaTeX: \displaystyle \left( \frac{836}{129}, \ \frac{4505}{129}\right)$ . Making a table gives to test the verticies in $LaTeX: \displaystyle p=17 x + 19 y$ gives

\begin{tabular}{|c|c|}\hline Point & Function \\[3pt] \hline $LaTeX: \displaystyle \left( 0, \ 0\right)$ & $LaTeX: \displaystyle 0$ \\[3pt] \hline $LaTeX: \displaystyle \left( 11, \ 0\right)$ & $LaTeX: \displaystyle 187$ \\[3pt] \hline $LaTeX: \displaystyle \left( \frac{836}{129}, \ \frac{4505}{129}\right)$ & $LaTeX: \displaystyle \frac{33269}{43}$ \\[3pt] \hline $LaTeX: \displaystyle \left( 0, \ 53\right)$ & $LaTeX: \displaystyle 1007$ \\[3pt] \hline \end{tabular} The gives the maximum value of $LaTeX: \displaystyle 1007$ located at $LaTeX: \displaystyle \left( 0, \ 53\right)$ .