Use the simplex method to maximize LaTeX:  \displaystyle p = 7 x + 18 y subject to LaTeX:  \displaystyle \begin{cases}86 x + 29 y \leq 2494 \\ 31 x + 62 y \leq 1922 \\x \geq 0, y \geq 0 \end{cases}

Adding the slack variables LaTeX:  \displaystyle s and LaTeX:  \displaystyle t to the inequalities gives:
LaTeX:  \begin{cases}86 x + 29 y +s = 2494 \\ 31 x + 62 y+t = 1922 \\ - 7 x - 18 y+p =0 \end{cases} This gives the first tableau:\begin{tabular}{|c|c|c|c|c|c|c|}\hline LaTeX:  \displaystyle  & LaTeX:  \displaystyle x & LaTeX:  \displaystyle y & LaTeX:  \displaystyle s & LaTeX:  \displaystyle t & LaTeX:  \displaystyle p & LaTeX:  \displaystyle  \\ \hline LaTeX:  \displaystyle s & LaTeX:  \displaystyle 86 & LaTeX:  \displaystyle 29 & LaTeX:  \displaystyle 1 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 2494 \\ \hline LaTeX:  \displaystyle t & LaTeX:  \displaystyle 31 & LaTeX:  \displaystyle 62 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 1 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 1922 \\ \hline LaTeX:  \displaystyle p & LaTeX:  \displaystyle -7 & LaTeX:  \displaystyle -18 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 1 & LaTeX:  \displaystyle 0 \\ \hline \end{tabular}
The pivot row is LaTeX:  \displaystyle t and the pivot column is LaTeX:  \displaystyle y . The departing variable is LaTeX:  \displaystyle t and the incoming variable is LaTeX:  \displaystyle y . Pivoting using the row operations:
LaTeX:  \displaystyle 62R_1-29R_2
LaTeX:  \displaystyle 31R_3+9R_2
\begin{tabular}{|c|c|c|c|c|c|c|}\hline LaTeX:  \displaystyle  & LaTeX:  \displaystyle x & LaTeX:  \displaystyle y & LaTeX:  \displaystyle s & LaTeX:  \displaystyle t & LaTeX:  \displaystyle p & LaTeX:  \displaystyle  \\ \hline LaTeX:  \displaystyle s & LaTeX:  \displaystyle 4433 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 62 & LaTeX:  \displaystyle -29 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 98890 \\ \hline LaTeX:  \displaystyle y & LaTeX:  \displaystyle 31 & LaTeX:  \displaystyle 62 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 1 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 1922 \\ \hline LaTeX:  \displaystyle p & LaTeX:  \displaystyle 62 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 9 & LaTeX:  \displaystyle 31 & LaTeX:  \displaystyle 17298 \\ \hline \end{tabular}
There are no negative values in row LaTeX:  \displaystyle p and this is the final tableau.The value of LaTeX:  \displaystyle s is LaTeX:  \displaystyle 1595 . The value of LaTeX:  \displaystyle y is LaTeX:  \displaystyle 31 . The max value is LaTeX:  \displaystyle p = 558