Solve the system by any method. \[ \begin{array}{c} x+y+z+w=0 \\ 3 x-y+3 z-w=0 \\ 3 x+5 y-z+5 w=-4 \\ x-2 y+3 z-w=0 \end{array} \]

Step 1 ：We are given a system of linear equations with four variables as follows:

Step 2 ：\[\begin{array}{c} x+y+z+w=0 \\ 3x-y+3z-w=0 \\ 3x+5y-z+5w=-4 \\ x-2y+3z-w=0 \end{array}\]

Step 3 ：We can represent this system in matrix form as follows:

Step 4 ：\[A = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 3 & -1 & 3 & -1 \\ 3 & 5 & -1 & 5 \\ 1 & -2 & 3 & -1 \end{bmatrix}, b = \begin{bmatrix} 0 \\ 0 \\ -4 \\ 0 \end{bmatrix}\]

Step 5 ：We solve this system to find the values of the variables x, y, z, and w.

Step 6 ：The solution to the system of equations is \(x = -1, y = 2, z = 1, w = -2\).

Step 7 ：Final Answer: \(\boxed{x = -1, y = 2, z = 1, w = -2}\)