2 A linear programming problem is Maximise \(\mathrm { P } = 2 \mathrm { x } - \mathrm { y } + \mathrm { z }\)
subject to
\(3 x - 4 y - z \leqslant 30\)
\(x - y \leqslant 6\)
\(x - 3 y + 2 z \geqslant - 2\)
and \(x \geqslant 0 , y \geqslant 0 , z \geqslant 0\)

1. Complete the table in the Printed Answer Booklet to represent the problem as an initial simplex tableau.
2. Carry out one iteration of the simplex algorithm.
3. State the values of \(x , y\) and \(z\) that result from your iteration. After two iterations the resulting tableau is

   \(P\)	\(x\)	\(y\)	\(z\)	\(s\)	\(t\)	\(u\)	RHS
   1	0	0	-2	0	2.5	0.5	16
   0	0	0	-2	1	-2.5	0.5	16
   0	1	0	-1	0	1.5	0.5	10
   0	0	1	-1	0	0.5	0.5	4

   The boundaries of the feasible region are planes, with edges each defined by two of \(x , y , z , s , t , u\) being zero.
   At each vertex of the feasible region there are three basic variables and three non-basic variables.
4. Interpret the second iteration geometrically by stating which edge of the feasible region is being moved along. As part of your geometrical interpretation, you should state the beginning vertex and end vertex of the second iteration.