SOLUTION: I've been working on this for hours, but keep taking the wrong path. I need to solve the 4x4 equations: 2w + x - y + 2z = 10 -w - 3x + y - 5z = -11 3w + 5x - 3y + z = 35 -4w -

2x1 + x2 - x3 + 2x4 = 10
-x1 - 3x2 + x3 - 5x4 = -11
3x1 + 5x2 - 3x3 + x4 = 35
-4x1 - 2x2 - 3x3 - x4 = 2
 
We rewrite the system of equations in matrix form and solve it using the Gauss method

2	1	-1	2	10
-1	-3	1	-5	-11
3	5	-3	1	35
-4	-2	-3	-1	2
 
Divide the 1st row by 2

1	0.5	-0.5	1	5
-1	-3	1	-5	-11
3	5	-3	1	35
-4	-2	-3	-1	2
 
to line 2 we add line 1 multiplied by 1; from line 3 we subtract line 1 multiplied by 3; 
to line 4 add line 1 multiplied by 4

1	0.5	-0.5	1	5
0	-2.5	0.5	-4	-6
0	3.5	-1.5	-2	20
0	0	-5	3	22
 
Divide the 2nd line by -2.5

1	0.5	-0.5	1	5
0	1	-0.2	1.6	2.4
0	3.5	-1.5	-2	20
0	0	-5	3	22
 
from line 1 we subtract line 2, multiplied by 0.5; from line 3 subtract line 2 multiplied by 3.5

1	0	-0.4	0.2	3.8
0	1	-0.2	1.6	2.4
0	0	-0.8	-7.6	11.6
0	0	-5	3	22
 
Divide the 3rd line by -0.8

1	0	-0.4	0.2	3.8
0	1	-0.2	1.6	2.4
0	0	1	9.5	-14.5
0	0	-5	3	22
 
to line 1 we add line 3 multiplied by 0.4; to line 2 add line 3 multiplied by 0.2;
to line 4 add line 3 multiplied by 5

1	0	0	4	-2
0	1	0	3.5	-0.5
0	0	1	9.5	-14.5
0	0	0	50.5	-50.5
 
Divide the 4th line by 50.5

1	0	0	4	-2
0	1	0	3.5	-0.5
0	0	1	9.5	-14.5
0	0	0	1	-1
 
from line 1 we subtract line 4, multiplied by 4; from line 2 we subtract line 4, multiplied by 3.5;
from line 3 subtract line 4 multiplied by 9.5

1	0	0	0	2
0	1	0	0	3
0	0	1	0	-5
0	0	0	1	-1
 
x1 = 2
x2 = 3
x3 = -5
x4 = -1

DO NOT bother to even look at that woman's RIDICULOUS method of solving this. As usual, it's complicated as HELL,
with all those solutions of variables that form UNNECESSARY fractions!! She needs to learn mathematics!

Looking at eqs (i) & (ii), it's obvious that "y" can be immediately eliminated by adding the two.
Likewise, looking at eqs (iii) & (iv), it's clear that "y" can also be immediately eliminated by subtracting one
equation from the other. So, let's do just that!

  2w +  x  - y + 2z =   10 ----- eq (i)
-  w - 3x  + y - 5z = - 11 ----- eq (ii)
   w - 2x - 3z = - 1 ---- Adding eqs (i) & (ii) ------ eq (v)

  3w + 5x - 3y +  z =   35 ----- eq (iii)
- 4w - 2x - 3y -  z =    2 ----- eq (iv)
  7w + 7x + 2z = 33 ----- Subtracting eq (iv) from eq (iii) ----- eq (vi)

- 3w - 9x  + 3y - 15z = - 33 ----- eq (ii) ---- Multiplying eq (ii) by 3 ---- eq (vii)
  3w + 5x - 3y +  z =   35 ----- eq (iii)
    - 4x - 14z = 2 ---- Adding eq (vii) & (iii)
    ------ Dividing by/Factoring out GCF, - 2 
       2x + 7z = - 1 ----- eq (viii)

 w - 2x - 3z = - 1 ---- eq (v)
7w + 7x + 2z = 33 ----- eq (vi)
7w - 14x  - 21z = - 7 ----- Multiplying eq (v) by 7 ---- eq (ix)
      21x + 23z = 40 ---- Subtracting eq (ix) from eq (vi) ------ eq (x)

Now, we have 2 equations in 2 UNKNOWNS:
      2x + 7z = - 1 ----- eq (viii)
      21x + 23z = 40 ---- eq (x)
      42x + 147z = - 21 ---- Multiplying eq (viii) by 21 ----- eq (xi)
      42x +  46z = 80 ------ Multiplying eq (x) by 2 ----- eq (xii)
            101z = - 101 --- Subtracting eq (xii) from eq (xi)
              

      2x + 7(- 1) = - 1 ---- Substituting - 1 for z in eq (viii)
      2x - 7 = - 1
      2x = 6
      

      w - 2(3) - 3(- 1) = - 1 ----- Substituting - 1 for z and 3 for x in eq (v)
      w - 6 + 3 = - 1
      w - 3 = - 1
      

      - 2 - 3(3)  + y - 5(- 1) = - 11 ----- Substituting - 1 for z, 3 for x, and 2 for w in eq (ii)
      - 2 - 9 + y + 5 = - 11
      - 6 + y = - 11