The point P$(x$, -1) lies on the segment joining A(2, 5) and B(8, -4).
Find the ratio AP : PB and the value of $x$.

Using $y:\frac{-4m+5n}{m+n}=-1$, so $-4m+5n=-m-n,6n=3m$ and $m:n=2$ : 1. Then $x=\frac{2\cdot 8+1\cdot 2}{3}=\frac{18}{3}=6$. The option 1 : 2 with $x=4$ inverts $m$ and $n$ consistently; 1 : 1 with $x=5$ assumes P is the midpoint, but the midpoint's $y$ is 0.5, not -1; the last option has the correct ratio but computes $x$ as $m{x}_1+n{x}_2$.