linear algebra
posted by DON .
solve using gauss jordan
10y − 4z + w = 1
x + 4y − z + w = 2
3x + 2y + z + 2w = 5
−2x − 8y + 2z − 2w = −4
x − 6y + 3z = 1

I see more equations (5) than unknowns (4). The problem could be overdetermined.
The last equation is a simple combination of the third and fourth. So there is a solution, but you don't need the x − 6y + 3z = 1 equation to get it.
For more about Gauss Jordan, see
http://aubuchon.jsc.vsc.edu/finiteNotes7SystemsUnique.htm