That's 4 equations in 4 variables

2w +  x +  y + 2z = -2
3w + 2x +  y -  z =  9
 w +  x +  y +  z =  0
 w + 2x + 2y -  z = 10

First we get it down to 3 equations in 3 variables
by choosing a variable to eliminate, and eliminating
it from 3 pairs of equations making sure all 4 equations
are used.  I will choose z to eliminate.

We eliminate z from the first two equations by
multiplying the 2nd eq. by 2 and adding to the 1st:

2w +  x +  y + 2z = -2
6w + 4x + 2y - 2z = 18
----------------------
8w + 5x + 3y      = 16

We eliminate z from the 2nd & 3rd equations by
just adding them as they are:

3w + 2x +  y -  z =  9
 w +  x +  y +  z =  0
----------------------
4w + 3x + 2y      =  9

We eliminate z from the 3rd & 4th equations by
just adding them as they are:


 w +  x +  y +  z =  0
 w + 2x + 2y -  z = 10
----------------------
2w + 3x + 3y      = 10

Now we have it down to 3 equations in 3 variables.

8w + 5x + 3y = 16
4w + 3x + 2y =  9
2w + 3x + 3y = 10

Next we get it down to 2 equations in 2 variables
by choosing a variable to eliminate, and eliminating
it from 2 pairs of equations, making sure that all
3 equations are used.  I choose w to eliminate:

We eliminate w from the 2nd & 3rd equations by multiplying
the second equation by -2 and adding it to the 1st equation:

 8w + 5x + 3y =  16
-8w - 6x - 4y = -18
-------------------
      -x -  y =  -2

We eliminate w from the first two equations by multiplying
the 3rd equation by -2 and adding it to the 2nd equation:

 4w + 3x + 2y =   9
-4w - 6x - 6y = -20
-------------------
     -3x - 4y = -11

Now we have it down to 2 equations in 2 variables.

      -x -  y =  -2
     -3x - 4y = -11

Next we get it down to 1 equation in 1 variables
by choosing a variable to eliminate.  I choose y 
to eliminate:
 
We eliminate y by multiplying the 1st equation by -4
and adding the 2nd equation to it:

      4x + 4y =   8
     -3x - 4y = -11
   ----------------
       x      =  -3

That's 1 equation in 1 variable solved. So we 
substitute in 

      -x - y = -2
   -(-3) - y = -2
       3 - y = -2
          -y = -5
           y =  5 

Then we substitute x = -3 and y = 5 in

     2w + 3x + 3y = 10
2w + 3(-3) + 3(5) = 10
      2w - 9 + 15 = 10
           2w + 6 = 10
               2w = 4
                w = 2 
Then we substitute x = -3 and y = 5 and w = 2 in

    w + x + y + z =  0
 2 + (-3) + 5 + z = 0
            4 + z = 0
                z = -4

So (w,x,y,z) = (2,-3,5,-4)

Edwin