Note: when you see a star symbol (*) it means multiply.

2w + 3y - 2z + x = 2.15 (1)
3x + 2z + y + w = 12.25 (2)
y + 2z - 3x + 3w = 10.35 (3)
5z + 2x + 4y + w = 22.90 (4)

(3)-(2): -6x + 2w = -1.90 (5)

2*(4) + 5*(1): 12w + 23y + 9x = 56.55 (6)

3*(5) + 2*(6): 30w + 46y = 107.4 (7)

2*(4) - 5*(3): 3y + 19x - 13w = -5.95 (8)

(2) + (1): 3w + 4x + 4y = 14.40 (9)

Using equations (6), (8) and (9):

3*(8) + 13*(9): 61y + 109x = 169.35 (10a)

(6) - 4*(9): 7y - 7x = -1.05 (10b)

Using equations (10a) and (10b):

7*(10a) - 61*(10b): 1190x = 1249.5
x = 1.05 (Divide both sides by 1190)

(From this point on, you can use any combination of equations)

Substitute x = 1.05 into (5): 2w = 4.4
w = 2.20 (Divide both sides by 2)

Substitute w = 2.20 into (7): 46y = 41.4
y = 0.9 (Divide both sides by 46)

Substitute w = 2.20, x = 1.05 and y = 0.9 into (1): -2z = -6
z = 3 (Divide both sides by -2)

Summary

w = 2.20
x = 1.05
y = 0.90
z = 3.00