5. The tableau below is the initial tableau for a three-variable linear programming problem in \(x , y\) and \(z\). The objective is to maximise the profit, \(P\).
| Basic variable | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value |
| \(r\) | 15 | - 2 | 3 | 1 | 0 | 0 | 180 |
| \(s\) | 10 | 1 | 1 | 0 | 1 | 0 | 80 |
| \(t\) | 1 | 6 | - 2 | 0 | 0 | 1 | 100 |
| \(P\) | - 1 | - 2 | - 5 | 0 | 0 | 0 | 0 |
- Using the information in the tableau, write down
- the objective function,
- the three constraints as inequalities.
- Taking the most negative number in the profit row to indicate the pivot column at each stage, solve this linear programming problem. Make your method clear by stating the row operations you use.
- State the final values of the objective function and each variable.