The line $2x+y=7$ crosses the segment joining A(-1, 3) and B(5, 9).
Find the coordinates of the point where it cuts AB.

Let the ratio be $m:n$, so the cutting point is $(\frac{5m-n}{m+n},\frac{9m+3n}{m+n})$. Substituting in $2x+y=7$ gives $2(5m-n)+(9m+3n)=7(m+n)$, so $19m+n=7m+7n,12m=6n$ and $m:n=1$ : 2. Hence $x=\frac{5-2}{3}=1$ and $y=\frac{9+6}{3}=5$, giving (1, 5); check $2(1)+5=7$. (3, 7) uses the swapped ratio 2 : 1 and does not satisfy the line; (2, 6) is the midpoint, assumed without testing the line; (1, 6) has the right $x$ but slips on $y$.