7. This question is to be answered on the sheet provided in the answer booklet. A chemical company makes 3 products \(X , Y\) and \(Z\). It wishes to maximise its profit \(\pounds P\). The manager considers the limitations on the raw materials available and models the situation with the following Linear Programming problem. Maximise $$\begin{gathered} P = 3 x + 6 y + 4 z
x \quad + \quad z \leq 4
x + 4 y + 2 z \leq 6
x + y + 2 z \leq 12
x \geq 0 , \quad y \geq 0 , \quad z \geq 0 \end{gathered}$$ subject to
where \(x , y\) and \(z\) are the weights, in kg , of products \(X , Y\) and \(Z\) respectively.
A possible initial tableau is

1. Explain
   1. the purpose of the variables \(r , s\) and \(t\),
   2. the final row of the tableau.
2. Solve this Linear Programming problem by using the Simplex alogorithm. Increase \(y\) for your first iteration and than increase \(x\) for your second iteration.
3. Interpret your solution.