Display the following linear programming problem in a Simplex tableau.
$$\begin{array} { l l }
\text { Maximise } & P = 4 x - 5 y + 6 z
\text { subject to } & 6 x + 7 y - 4 z \leqslant 30
& 2 x + 4 y - 5 z \leqslant 8
& x \geqslant 0 , y \geqslant 0 , z \geqslant 0
\end{array}$$
The Simplex method is to be used to solve this problem.
Explain why it is not possible to choose a pivot from the \(z\)-column initially.
Identify the initial pivot and explain why this particular element should be chosen.
Perform one iteration using your initial tableau from part (a).
State the values of \(x , y\) and \(z\) after this first iteration.
Without performing any further iterations, explain why \(P\) has no finite maximum value.
Using the same inequalities as in part (a), the problem is modified to
$$\text { Maximise } \quad Q = 4 x - 5 y - 20 z$$
Write down a modified initial tableau and the revised tableau after one iteration.