5. A linear programming problem in \(x\) and \(y\) is described as follows. $$\begin{array} { l l } \text { Maximise } & \mathrm { P } = 5 x + 3 y
\text { subject to: } & x \geqslant 3
& x + y \leqslant 9
& 15 x + 22 y \leqslant 165
& 26 x - 50 y \leqslant 325 \end{array}$$

1. Add lines and shading to Diagram 1 in the answer book to represent these constraints. Hence determine the feasible region and label it R .
2. Use the objective line method to find the optimal vertex, V, of the feasible region. You must draw and label your objective line and label vertex V clearly.
3. Calculate the exact coordinates of vertex V and hence calculate the corresponding value of P at V . The objective is now to minimise \(5 x + 3 y\), where \(x\) and \(y\) are integers.
4. Write down the minimum value of \(5 x + 3 y\) and the corresponding value of \(x\) and corresponding value of \(y\).