Display the following linear programming problem in a Simplex tableau.
$$\begin{array} { l l }
\text { Maximise } & P = 5 x + 8 y + 7 z
\text { subject to } & 3 x + 2 y + z \leqslant 12
& 2 x + 4 y + 5 z \leqslant 16
& x \geqslant 0 , y \geqslant 0 , z \geqslant 0
\end{array}$$
The Simplex method is to be used by initially choosing a value in the \(y\)-column as a pivot.
Explain why the initial pivot is 4 .
Perform two iterations of your tableau from part (a) using the Simplex method.
State the values of \(P , x , y\) and \(z\) after your second iteration.
State, giving a reason, whether the maximum value of \(P\) has been achieved.