10. While solving a maximizing linear programming problem, the following tableau was obtained.
| \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value |
| \(r\) | 0 | 0 | \(1 \frac { 2 } { 3 }\) | 1 | 0 | \(- \frac { 1 } { 6 }\) | \(\frac { 2 } { 3 }\) |
| \(y\) | 0 | 1 | \(3 \frac { 1 } { 3 }\) | 0 | 1 | \(- \frac { 1 } { 3 }\) | \(\frac { 1 } { 3 }\) |
| \(x\) | 1 | 0 | - 3 | 0 | - 1 | \(\frac { 1 } { 2 }\) | 1 |
| \(P\) | 0 | 0 | 1 | 0 | 1 | 1 | 11 |
- Explain why this is an optimal tableau.
- Write down the optimal solution of this problem, stating the value of every variable.
- Write down the profit equation from the tableau. Use it to explain why changing the value of any of the non-basic variables will decrease the value of \(P\).