How to solve the following question using Laplace Transform?

x' = x + z
y' = x + y
z' = -2x - z
x(0)=1, y(0)=0, z(0)=1

Laplace Transform
sX(s)-1=X(s)+Z(s)
sY(s)-0=X(s)+Y(s)
sZ(s)-1=-2X(s)-Z(s)

(s-1)X(s)-Z(s)=1 ...(1)
X(s)-(s-1)Y(s)=0 ...(2)
2X(s)+(s+1)Z(s)=1 ...(3)
(1)*(s+1) + (3)
(s^2-1+2)X(s)=s+1+1
X(s)=(s+2)/(s^2+1) ...(4)
x(t) = cos t + 2 sin t
sub (4) into (2)
(s+2)/(s^2+1)-(s-1)Y(s)=0
Y(s)=(s+2)/[(s-1)(s^2+1)]
set Y(s)= A/(s-1) + (Bs+C)/(s^2+1)
A(s^2+1)+(s-1)(Bs+C)=s+2
A+B=0
C-B=1
A-C=2
C = - 1/2, B = - 3/2, A = 3/2
Y(s)= A/(s-1) + (Bs+C)/(s^2+1)
y(t) = (3/2) e^t - (3/2) cos t -(1/2) sin t
sub (4) into (1)
(s-1)(s+2)/(s^2+1)-Z(s)=1
(s^2+s-2)/(s^2+1)-Z(s)=1
Z(s)=(s-3)/(s^2+1)
z(t) = cos t - 3 sin t

Ans: x(t) = cos t + 2 sin t
y(t) = (3/2) e^t - (3/2) cos t - (1/2) sin t
z(t) = cos t - 3 sin t