Use laplace transform to solve the given system of differential equations

dx/dt=x-2y
and
dy/dt=5x-y
x(0)=-1 & y(0)=2

I took the laplace of both equations and got
sL[x]+1=L[x]-2L[y]
sL[y]-2=5L[x]-L[y]

I don't know how to simplify in terms of L[x]= & L[y]=

Rewrite as
(s - 1) L[x] + 2 L[y] = -1
5 L[x] + (-s - 1) L[y] = -2

Multiply top equation by 5, the bottom equation by (s - 1), and subtract:
5(s - 1) L[x] + 10 L[y] = -5
- [5(s - 1) L[x] + (-s - 1)(s - 1) L[y] = -2(s - 1)]
--------------------------------------…
(10 + (s^2 - 1)) L[y] = -5 + 2(s - 1)
==> L[y] = (2s - 7)/(s^2 + 9).

Multiply top equation by (s+1), the bottom equation by 2, and add:
(s - 1)(s+1) L[x] + 2(s+1) L[y] = -(s+1)
+ [10 L[x] + 2(-s - 1) L[y] = -4]
--------------------------------------…
(s^2 + 9) L[x] = -s - 5
==> L[x] = (-s - 5)/(s^2 + 9)

Inverting yields
x(t) = -cos(3t) - (5/3) sin(3t)
y(t) = 2 cos(3t) - (7/3) sin(3t).

I hope this helps!