Maximize LaTeX: \displaystyle p = 3 x + 7 y subject to LaTeX: \displaystyle \begin{cases}100 x + 5 y \leq 500 \\ 96 x + 65 y \leq 6240 \\ 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{65}{301}, \ \frac{28800}{301}\right) . Making a table gives to test the verticies in LaTeX: \displaystyle p=3 x + 7 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( 5, \ 0\right) & LaTeX: \displaystyle 15 \\[3pt] \hline LaTeX: \displaystyle \left( \frac{65}{301}, \ \frac{28800}{301}\right) & LaTeX: \displaystyle \frac{201795}{301} \\[3pt] \hline LaTeX: \displaystyle \left( 0, \ 96\right) & LaTeX: \displaystyle 672 \\[3pt] \hline \end{tabular}
The gives the maximum value of LaTeX: \displaystyle 672 located at LaTeX: \displaystyle \left( 0, \ 96\right) .