5. A linear programming problem in \(x , y\) and \(z\) is described as follows. Maximise \(P = 2 x + 3 y + z\)
subject to \(\quad 2 y - 3 z \leqslant 30\) $$\begin{array} { r } - 3 x + y + z \leqslant 60
x + 4 y - z \leqslant 80 \end{array}$$

1. Complete the initial tableau in the answer book for this linear programming problem.
   (3)
2. Taking the most negative number in the profit row to indicate the pivot column, perform one complete iteration of the simplex algorithm to obtain a new tableau, T. Make your method clear by stating the row operations you use.
   (5)
3. Write down the profit equation given by T and state the values of the slack variables given by T . The following tableau is obtained after further iterations.

   Basic variable	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	\(t\)	Value
   \(r\)	0	2	-3	1	0	0	30
   \(s\)	0	13	-2	0	1	3	300
   \(x\)	1	4	-1	0	0	1	80
   \(P\)	0	5	-3	0	0	2	160

4. Explain why no optimal solution can be found by applying the simplex algorithm to the above tableau.