x + y + z - 2w = -4
2y + z + 3w = 4
2x + y - z + 2w = 5
x - y + w = 4
z is already eliminated in the 4th equation,
Add the 1st and 3rd equation to eliminate z
x + y + z - 2w = -4
2x + y - z + 2w = 5
---------------------
3x + 2y = 1
Add the 2nd and 3rd equation to eliminate z
2y + z + 3w = 4
2x + y - z + 2w = 5
---------------------
2x + 3y + 5w = 9
So now we have this system of equations
in only x, y, and w. Not z:
x - y + w = 4
3x + 2y = 1
2x + 3y + 5w = 9
w is already eliminated from the 2nd
equation, so multiply the 1st equation
by -5 and add it to the 3rd equation:
-5x + 5y - 5w = -20
2x + 3y + 5w = 9
-------------------
-3x + 8y = -11
So now we have this system of equations
in only x and y. Not z or w:
3x + 2y = 1
-3x + 8y = -11
----------------
10y = -10
y = -1
Substitute y = -1 in
3x + 2y = 1
3x + 2(-1) = 1
3x - 2 = 1
3x = 3
x = 1
Substitute x = 1 and y = -1 in
x - y + w = 4
1 - (-1) + w = 4
1 + 1 + w = 4
2 + w = 4
w = 2
Substitute x = 1, y = -1, and w = 2 in
x + y + z - 2w = -4
1 + (-1) + z - 2(2) = -4
1 - 1 + z - 4 = -4
z - 4 = -4
z = 0
(x,y,z,w) = (1,-1,0,2)
Edwin
