7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{48e785c0-7de5-450f-862c-4dd4d169adf9-10_993_1268_221_402}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
Figure 3 shows the constraints of a linear programming problem in \(x\) and \(y\), where \(R\) is the feasible region. The equations of two of the lines have been shown in Figure 3.
Given that \(k\) is a positive constant,
- determine, in terms of \(k\) where necessary, the inequalities that define \(R\).
The objective is to maximise \(P = 5 x + k y\)
Given that the value of \(P\) is 38 at the optimal vertex of \(R\), - determine the possible value(s) of \(k\). You must show algebraic working and make your method clear.
(Total 11 marks)