4 Consider the linear programming problem: $$\begin{array} { l l } \text { maximise } & P = 3 x - 5 y ,
\text { subject to } & x + 5 y \leqslant 12 ,
& x - 5 y \leqslant 10 ,
& 3 x + 10 y \leqslant 45 ,
\text { and } & x \geqslant 0 , y \geqslant 0 . \end{array}$$

1. Represent the problem as an initial Simplex tableau.
2. Identify the entry on which to pivot for the first iteration of the Simplex algorithm. Explain how you made your choice of column and row.
3. Perform oneiteration of the Simplex algorithm. Write down the values of \(\mathrm { x } , \mathrm { y }\) and P after this iteration.
4. Show that \(\mathrm { x } = 11 , \mathrm { y } = 0.2\) is a feasible solution and that it gives a bigger value of P than that in part (iii).