7. \begin{figure}[h] \end{figure} Figure 5 shows the constraints of a linear programming problem in \(x\) and \(y\) where \(R\) is the feasible region. The objective is to maximise \(P = x + k y\), where \(k\) is a positive constant.
The optimal vertex of \(R\) is to be found using the Simplex algorithm.

1. Set up an initial tableau for solving this linear programming problem using the Simplex algorithm. After two iterations of the Simplex algorithm a possible tableau \(T\) is

   b.v.	\(x\)	\(y\)	\(S _ { 1 }\)	\(s _ { 2 }\)	\(S _ { 3 }\)	\(s _ { 4 }\)	Value
   \(s _ { 1 }\)	0	0	1	\(- \frac { 3 } { 5 }\)	0	\(\frac { 1 } { 5 }\)	1
   \(x\)	1	0	0	\(\frac { 1 } { 5 }\)	0	\(- \frac { 2 } { 5 }\)	2
   \(S _ { 3 }\)	0	0	0	\(- \frac { 11 } { 5 }\)	1	\(\frac { 12 } { 5 }\)	22
   \(y\)	0	1	0	\(\frac { 2 } { 5 }\)	0	\(\frac { 1 } { 5 }\)	5
   \(P\)	0	0	0	\(\frac { 1 } { 5 } + \frac { 2 } { 5 } k\)	0	\(- \frac { 2 } { 5 } + \frac { 1 } { 5 } k\)	\(5 k + 2\)

2. State the value of each variable after the second iteration.
   (1) It is given that \(T\) does not give an optimal solution to the linear programming problem.
   After a third iteration of the Simplex algorithm the resulting tableau does give an optimal solution to the problem.
3. Perform the third iteration of the Simplex algorithm and hence determine the range of possible values for \(P\). You should state the row operations you use and make your method and working clear.
   (9)