CurveGuy is correct about the method but assumed the initial conditions are all 0. For example [MATH]\dot X = sX-x(0)[/MATH]. This leads to the linear system:
[MATH]\begin{pmatrix} sX-x(0)\\sY-y(0)\\sZ-z(0) \end{pmatrix} =
\begin{pmatrix} 5 & 5 & 2 \\ -6 & -6 & -5 \\ 6 & 6 & 5 \end{pmatrix} =
\begin{pmatrix} X \\ Y \\ Z \end{pmatrix} [/math]where the initial conditions might not be [MATH]0[/MATH]. You can solve the system by any favorite method you choose to get the transforms [MATH]X,~Y,~Z[/MATH] after which you need to find their inverses.
That [MATH]6[/MATH] with a line above it should be [MATH]-6[/MATH]. There appears to be a problem with Latex rendering.