6 A linear programming problem is
Maximise \(\mathrm { P } = 2 \mathrm { x } - \mathrm { y }\)
subject to $$\begin{aligned} 3 x + y - 4 z & \leqslant 24
5 x - 3 z & \leqslant 60
- x + 2 y + 3 z & \leqslant 12 \end{aligned}$$ and \(x \geqslant 0 , y \geqslant 0 , z \geqslant 0\)

1. 1. Represent this problem as an initial simplex tableau.
   2. Carry out one iteration of the simplex algorithm. After two iterations the resulting tableau is

      \(P\)	\(x\)	\(y\)	\(z\)	\(s\)	\(t\)	\(u\)	RHS
      1	0	\(\frac { 5 } { 11 }\)	0	\(- \frac { 6 } { 11 }\)	\(\frac { 8 } { 11 }\)	0	\(30 \frac { 6 } { 11 }\)
      0	1	\(- \frac { 3 } { 11 }\)	0	\(- \frac { 3 } { 11 }\)	\(\frac { 4 } { 11 }\)	0	\(15 \frac { 3 } { 11 }\)
      0	0	\(- \frac { 5 } { 11 }\)	1	\(- \frac { 5 } { 11 }\)	\(\frac { 3 } { 11 }\)	0	\(5 \frac { 5 } { 11 }\)
      0	0	\(\frac { 34 } { 11 }\)	0	\(\frac { 12 } { 11 }\)	\(- \frac { 5 } { 11 }\)	1	\(10 \frac { 10 } { 11 }\)

2. 1. Write down the basic variables after two iterations.
   2. Write down the exact values of the basic feasible solution for \(x , y\) and \(z\) after two iterations.
   3. State what you can deduce about the optimal value of the objective for the original problem. You are now given that, in addition to the constraints above, \(\mathrm { x } + \mathrm { y } + \mathrm { z } = 9\).
3. Use the additional constraint to rewrite the original constraints in terms of \(x\) and \(y\) but not \(z\).
4. Explain why the simplex algorithm cannot be applied to this new problem without some modification.