need work please!

x+2y+2z-w=0
x-y+3w=7
2x-y+z+w=6
2x+2y+3z-3w=-1

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I would solve this using elimination via Gaussian elimination.

Turn it into a matrix and reduce it using reduced row echelon form.

\begin{bmatrix}1&2&1&-1&0\\1&-1&3&0&7\\2&-1&1&1&6\\2&2&3&-3&-1\end{bmatrix}

When you get it into this form, you have your solutions:

\begin{bmatrix}1&0&0&0&x\\0&1&0&0&y\\0&0&1&0&z\\0&0&0&1&w\end{bmatrix}

Your solutions are in the right most column.

I'm totally stuck. I've been trying to cancel out things and just wind up cancelling out the whole equation, unable to find what any letters equal

This is tedious stuff. In my opinion, these should be done with a calculator.

I am sorry, but I am not going to post all this step by step.

Here is a screen capture from my TI-92. Now, at least, you can see what you're shooting for.

the answer is
x = 16
y = 9
z =-17
w = 0 and the row manipulation is as follows

r2-r1
r3-r4
r4-2r1
r2-2r4
r3+3r2
r4+2r2
r3/-2
r4+r3
r1-2r2
r1-2r3
r1-r4
r3+2r4

By the way I used gauss jordan

btw i used gauss jordan

Let's just hope the OP hasn't been waiting around for over a year to turn in his homework!