(1) 4w + x + 2y - 3z = -16
(2) -3w + 3x - y + 4z = 20
(3) -w + 2x + 5y + z = -4
(4) 5w + 4x + 3y - z = -10
Pick a letter to eliminate. I pick z.
Add equations (3) and (4) as they are
(3) -w + 2x + 5y + z = -4
(4) 5w + 4x + 3y - z = -10
--------------------------------
4w + 6x + 8y = -14
That can be simplified by dividing through by 2
(5) 2w + 3x + 4y = -7
Multiply (4) by 4 [to add to (2) to eliminate z]
20w + 16x + 12y - 4z = -40
(2) -3w + 3x - y + 4z = 20
----------------------------------
(6) 17w + 19x + 11y = -20
Multiply (3) by 3 [to add to (1) to eliminate z]
-3w + 6x + 15y + 3z = -12
(1) 4w + x + 2y - 3z = -16
----------------------------------
(7) w + 7x + 17y = -28
Now we have reduced the system to 3 equations in 3 unknowns:
(5) 2w + 3x + 4y = -7
(6) 17w + 19x + 11y = -20
(7) w + 7x + 17y = -28
Pick a letter to eliminate. I pick w.
Multiply (7) by -2 [to add to (5) to eliminate w]
-2w - 14x - 34y = 56
(5) 2w + 3x + 4y = -7
------------------------------
(8) -11x - 30y = 49
Multiply (7) by -17 [to add to (6) to eliminate w]
-17w - 119x - 289y = 476
(6) 17w + 19x + 11y = -20
--------------------------------
-100x - 278y = 456
That can be simplified by dividing through by 2
(9) -50x - 139y = 228
Now we have reduced the system to 2 equations in 2 unknowns:
(8) -11x - 30y = 49
(9) -50x - 139y = 228
Pick a letter to eliminate. I pick x.
the least common multiple of 11 and 50 is 550. So we multiply
(8) by 50, and(9) by -11 so that the x's will cancel:
-550x - 1500y = 2450
550x + 1529y = -2508
-------------------------
29y = -58
(10) y = -2
Substitute in (8)
-11x - 30(-2) = 49
-11x + 60 = 49
-11x = -11
(11) x = 1
Substitute x = 1 and y = -2 in (5)
2w + 3x + 4y = -7
2w + 3(1) + 4(-2) = -7
2w + 3 - 8 = -7
2w - 5 = -7
2w = -2
w = -1
Substitute x = 1, y = -2 and w = -1 in (3)
-(-1) + 2(1) + 5(-2) + z = -4
1 + 2 - 10 + z = -4
-7 + z = -4
z = 3
(w,x,y,z) = (-1,1,-2,3)
Edwin