4 A linear programming problem involving variables $x$ and $y$ is to be solved. The objective function to be maximised is $P = 3 x + 5 y$. The initial Simplex tableau is given below.

\begin{center}
\begin{tabular}{ | c | r | r | c | c | c | c | }
\hline
$\boldsymbol { P }$ & $\boldsymbol { x }$ & $\boldsymbol { y }$ & $\boldsymbol { s }$ & $\boldsymbol { t }$ & $\boldsymbol { u }$ & value \\
\hline
1 & - 3 & - 5 & 0 & 0 & 0 & 0 \\
\hline
0 & 1 & 2 & 1 & 0 & 0 & 36 \\
\hline
0 & 1 & 1 & 0 & 1 & 0 & 20 \\
\hline
0 & 4 & 1 & 0 & 0 & 1 & 39 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item In addition to $x \geqslant 0 , y \geqslant 0$, write down three inequalities involving $x$ and $y$ for this problem.
\item \begin{enumerate}[label=(\roman*)]
\item By choosing the first pivot from the $\boldsymbol { y }$-column, perform one iteration of the Simplex method.
\item Explain how you know that the optimal value has not been reached.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Perform one further iteration.
\item Interpret the final tableau and state the values of the slack variables.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA D2 2007 Q4 [14]}}