Maximize LaTeX: \displaystyle p = 17 x + 19 y subject to LaTeX: \displaystyle \begin{cases}81 x + 7 y \leq 567 \\ 15 x + 78 y \leq 1170 \\ 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{12012}{2071}, \ \frac{28755}{2071}\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( 7, \ 0\right) & LaTeX: \displaystyle 119 \\[3pt] \hline LaTeX: \displaystyle \left( \frac{12012}{2071}, \ \frac{28755}{2071}\right) & LaTeX: \displaystyle \frac{750549}{2071} \\[3pt] \hline LaTeX: \displaystyle \left( 0, \ 15\right) & LaTeX: \displaystyle 285 \\[3pt] \hline \end{tabular}
The gives the maximum value of LaTeX: \displaystyle \frac{750549}{2071} located at LaTeX: \displaystyle \left( \frac{12012}{2071}, \ \frac{28755}{2071}\right) .