Use the simplex method to maximize LaTeX:  \displaystyle p = 14 x + 9 y subject to LaTeX:  \displaystyle \begin{cases}61 x + 43 y \leq 2623 \\ 9 x + 63 y \leq 567 \\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}61 x + 43 y +s = 2623 \\ 9 x + 63 y+t = 567 \\ - 14 x - 9 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 61 & LaTeX:  \displaystyle 43 & LaTeX:  \displaystyle 1 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 2623 \\ \hline LaTeX:  \displaystyle t & LaTeX:  \displaystyle 9 & LaTeX:  \displaystyle 63 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 1 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 567 \\ \hline LaTeX:  \displaystyle p & LaTeX:  \displaystyle -14 & LaTeX:  \displaystyle -9 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 1 & LaTeX:  \displaystyle 0 \\ \hline \end{tabular}
The pivot row is LaTeX:  \displaystyle s and the pivot column is LaTeX:  \displaystyle x . The departing variable is LaTeX:  \displaystyle s and the incoming variable is LaTeX:  \displaystyle x . Pivoting using the row operations:
LaTeX:  \displaystyle 61R_2-9R_1
LaTeX:  \displaystyle 61R_3+14R_1
\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 x & LaTeX:  \displaystyle 61 & LaTeX:  \displaystyle 43 & LaTeX:  \displaystyle 1 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 2623 \\ \hline LaTeX:  \displaystyle t & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 3456 & LaTeX:  \displaystyle -9 & LaTeX:  \displaystyle 61 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 10980 \\ \hline LaTeX:  \displaystyle p & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 53 & LaTeX:  \displaystyle 14 & LaTeX:  \displaystyle 0 & LaTeX:  \displaystyle 61 & LaTeX:  \displaystyle 36722 \\ \hline \end{tabular}
There are no negative values in row LaTeX:  \displaystyle p and this is the final tableau.The value of LaTeX:  \displaystyle x is LaTeX:  \displaystyle 43 . The value of LaTeX:  \displaystyle t is LaTeX:  \displaystyle 180 . The max value is LaTeX:  \displaystyle p = 602