You can put this solution on YOUR website! The inverse of variables is turning you up side down! Fix it!
Merely let a = 1/x, b = 1/y and c = 1/z and yield the friendly 3x3 system of equations
2a + 3b - 2c = -1
3a - 18b + 5c = 4
5a + 3b - c = 1
Use Mathcad or your handy dandy TI-84 to get a, b and c. Then just get their inverse to get x, y, and z. You can also solve "by hand" using the inverse of a 3x3 matrix.