Where are you stuck? It would help if you showed your working.

Start with notation .... let LT[x(t)] = X(s) and LT[y(t)] = Y(s), say.

Now take take the Laplace transform of each differential equation.

This gives two linear equations in X(s) and Y(s) that need to be solved simultaneously.

Take the inverse Laplace transform of each resulting solution to get x(t) and y(t).