The points $(5,1,1)$, $(-1,4,-8)$, and $(4,4,-3)$ lie on a unique plane. Use linear algebra to find the equation of the plane and then determine where the line crosses the $y$-axis.

Equation of plane (use $x$, $y$, and $z$ as the variables):
Crosses the $y$-axis at the point: $\bigg($ , , $\bigg)$