3.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{049de386-42a9-4f16-8be3-9324382e4988-04_1684_1492_194_283}
\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 given.
- Determine the inequalities that define the feasible region.
- Find the exact coordinates of the vertices of the feasible region.
The objective is to maximise \(P\), where \(P = k x + y\).
- For the case \(k = 2\), use point testing to find the optimal vertex of the feasible region.
- For the case \(k = 2.5\), find the set of points for which \(P\) takes its maximum value.