6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{87f0e571-e708-4ca9-adc7-4ed18e144d32-07_1502_1659_230_210}
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\caption{Figure 4}
\end{figure}
Figure 4 shows the constraints of a linear programming problem in \(x\) and \(y\), where \(R\) is the feasible region.
The vertices of the feasible region are \(A ( 4,7 ) , B ( 5,3 ) , C ( - 1,5 )\) and \(D ( - 2,1 )\).
- Determine the inequality that defines the boundary of \(R\) that passes through vertices \(A\) and \(C\), leaving your answer with integer coefficients only.
The objective is to maximise \(P = 5 x + y\)
- Find the coordinates of the optimal vertex and the corresponding value of \(P\).
The objective is changed to maximise \(Q = k x + y\)
- If \(k\) can take any value, find the range of values of \(k\) for which \(A\) is the only optimal vertex.