7 A linear programming problem is
Maximise \(P = 4 x + y\)
subject to
$$\begin{aligned}
3 x - y & \leqslant 30
x + y & \leqslant 15
x - 3 y & \leqslant 6
\end{aligned}$$
and \(x \geqslant 0 , y \geqslant 0\)
- Use a graphical method to find the optimal value of \(P\), and the corresponding values of \(x\) and \(y\).
An additional constraint is introduced.
This constraint means that the value of \(y\) must be at least \(k\) times the value of \(x\), where \(k\) is a positive constant. - Determine the set of values of \(k\) for which the optimal value of \(P\) found in part (a) is unchanged.
- Determine, in terms of \(k\), the values of \(x , y\) and \(P\) in the cases when the optimal solution is different from that found in part (a).