Vectors in Space · Medium problem

Two independent thrusters push a satellite: firing thruster $A$ for $x$ seconds contributes a displacement of $x\mathbf a$, and firing thruster $B$ for $y$ seconds contributes $y\mathbf b$, where $\mathbf a = (1,2,-1)$ and $\mathbf b = (2,-1,3)$ (components in km, one km per second of firing). Mission control needs a net displacement of exactly $8$ km in the $x$-direction and $1$ km in the $y$-direction. Firing both thrusters for the needed durations also produces some displacement $k$ km in the $z$-direction, whether mission control wants it or not. Find $k$.