Oh, then I'll show you why there isn't one then.
When doing the simple version of Gaussian elimination, always look for "bare" variables first (variables that has a coefficient of 1, otherwise known as "that variable with no numbers"). In this case:
3x - 2y + 2z - w = 2 (1)
4x + y + z + 6w = 8 (2)
-3x + 2y - 2z + w = 5 (3)
5x + 3z - 2w = 1 (4)
In this case, we notice that in Equation 1 and 3, both w are bare (although there's a minus sign before the w in Eq. 1). Shift w around in Eq. 1 to make it positive again. Now change the subject:
3x - 2y + 2z - w = 2 becomes w = 3x - 2y + 2z - 2 and
-3x + 2y - 2z + w = 5 becomes w = 5 + 3x - 2y + 2z. Now that w is (sort of) found, hook them together:
3x - 2y + 2z - 2 = 5 + 3x - 2y + 2z. You see that after simplifying you're left with -2 = 5, which is obviously impossible.
Therefore, there is no solution since two of the so-called "related equations" contradict each other.