$\begin{array}{l}3 x-5 y+z=9 \\ x-3 y-2 z=-8 \\ 5 x-6 y+3 z=15\end{array}$

Step 1 ：We are given the system of linear equations: \[\begin{align*}3x - 5y + z &= 9, \x - 3y - 2z &= -8, \5x - 6y + 3z &= 15.\end{align*}\]

Step 2 ：We can represent this system in matrix form as follows: \[A = \begin{bmatrix}3 & -5 & 1 \1 & -3 & -2 \5 & -6 & 3\end{bmatrix}, b = \begin{bmatrix}9 \-8 \15\end{bmatrix}\]

Step 3 ：We solve the system by finding the inverse of matrix A and multiplying it with matrix b. The solution is given by: \[\begin{bmatrix}x \y \z\end{bmatrix} = A^{-1}b\]

Step 4 ：The solution to the system of equations is \(x = -6\), \(y = -4\), and \(z = 7\).

Step 5 ：Final Answer: \(\boxed{x = -6, y = -4, z = 7}\)