To solve this system of equations, we can write the system in matrix form AX = B, where A is the matrix of coefficients, X is the column vector of variables, and B is the column vector of constants.

The matrix A is:
[[1, 2, 3, 4],
[2, 1, 2, 3],
[3, 2, 1, 2],
[4, 3, 2, 1]]

The column vector X is:
[x1,
x2,
x3,
x4]

The column vector B is:
[5,
1,
1,
-5]

Therefore, the augmented matrix [A|B] is:
[[1, 2, 3, 4 | 5],
[2, 1, 2, 3 | 1],
[3, 2, 1, 2 | 1],
[4, 3, 2, 1 | -5]]

Performing row operations to row reduce the augmented matrix to row-echelon form, we get:
[[1, 0, 0, 0 | 8],
[0, 1, 0, 0 | -12],
[0, 0, 1, 0 | 2],
[0, 0, 0, 1 | 15]]

Therefore, the solution to the system of equations is:
x1 = 8
x2 = -12
x3 = 2
x4 = 15