Answer: \(\dfrac{AX}{XY} = \dfrac{3}{2}\)
In \(\triangle ABY\), BX bisects \(\angle B\), so by the angle bisector theorem
\[\frac{AX}{XY} = \frac{AB}{BY}\]In \(\triangle ACY\), CX bisects \(\angle C\), so
\[\frac{AX}{XY} = \frac{AC}{CY}\]Hence \(\frac{AB}{BY} = \frac{AC}{CY} = \frac{AX}{XY}\). By the property of equal ratios,
\[\frac{AX}{XY} = \frac{AB + AC}{BY + CY} = \frac{5 + 4}{6} = \frac{3}{2}\](BY + CY = BC = 6.)