Problem

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,

(a) determine, in terms of $k$ where necessary, the inequalities that define $R$.

The objective is to maximise $P = 5x + ky$.

Given that the value of $P$ is 38 at the optimal vertex of $R$,

(b) determine the possible value(s) of $k$. You must show algebraic working and make your method clear.