It would seem ill advised to me to make this a second order system considering you're only given two initial conditions(ICs). Basically anytime you have a DE when you solve them you're always doing some type of integration in order to solve them and in order to fully solve a DE you need the same number of initial conditions as the highest order derivative in the DE. In this case you are give 2 first order equations with 2 initial conditions so the system is solvable; however, in your first step you turn the system into a second order system of 2 equations so now you would need 4 initial conditions in order to solve it. Since you're not given 4 ICs you should keep this a first order system.

You're on the right track as far as taking the Laplace Transform of both equations. Take the Laplace Transform of the above system and solve it for $x(t)$ and $y(t)$.