Display the following linear programming problem in a Simplex tableau.
$$\begin{array} { l c }
\text { Maximise } & P = 3 x + 2 y + 4 z
\text { subject to } & x + 4 y + 2 z \leqslant 8
& 2 x + 7 y + 3 z \leqslant 21
& x \geqslant 0 , y \geqslant 0 , z \geqslant 0
\end{array}$$
Use the Simplex method to perform one iteration of your tableau for part (a), choosing a value in the \(z\)-column as pivot.
Perform one further iteration.
State whether or not this is the optimal solution, and give a reason for your answer.