5 A linear programming problem involving variables \(x\) and \(y\) is to be solved. The objective function to be maximised is \(P = 4 x + 9 y\). The initial Simplex tableau is given below.
| \(\boldsymbol { P }\) | \(\boldsymbol { x }\) | \(\boldsymbol { y }\) | \(r\) | \(s\) | \(\boldsymbol { t }\) | value |
| 1 | -4 | -9 | 0 | 0 | 0 | 0 |
| 0 | 3 | 7 | 1 | 0 | 0 | 33 |
| 0 | 1 | 2 | 0 | 1 | 0 | 10 |
| 0 | 2 | 7 | 0 | 0 | 1 | 26 |
- Write down the three inequalities in \(x\) and \(y\) represented by this tableau.
- The Simplex method is to be used to solve this linear programming problem by initially choosing a value in the \(x\)-column as the pivot.
- Explain why the initial pivot has value 1.
- Perform two iterations using the Simplex method.
- Comment on how you know that the optimum solution has been achieved and state your final values of \(P , x\) and \(y\).