7.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{27296f39-bd03-47ff-9a5e-c2212d0c68ed-09_956_1290_212_383}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{center}
\end{figure}

Figure 5 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 and the three intersection points, $A$, $B$ and $C$, are shown. The coordinates of $C$ are $\left( \frac { 35 } { 4 } , \frac { 15 } { 4 } \right)$

The objective function is $P = x + 3 y$\\
When the objective is to maximise $x + 3 y$, the value of $P$ is 24\\
When the objective is to minimise $x + 3 y$, the value of $P$ is 10
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find the coordinates of $A$ and $B$.
\item Determine the inequalities that define $R$.

An additional constraint, $y \geqslant k x$, where $k$ is a positive constant, is added to the linear programming problem.
\end{enumerate}\item Determine the greatest value of $k$ for which this additional constraint does not affect the feasible region.
\end{enumerate}

\hfill \mbox{\textit{Edexcel D1 2022 Q7 [10]}}