There are many ways to solve it. You didn't specify which way.
I arbitrarily picked elimination. Which method are you studying?
Gaussian elimination? Cramer's rule? If you're studying another
method, then post again telling what method. We aren't mind
readers.
x + y + z - w = 2
3x + y - z + w = 8
x - 5y + 2z + w = 1
-x - y + z + 3w = 4
1. Pick a letter to eliminate. I'll pick w
2. Pick two equations to eliminate it from. I pick the eq. 1 and eq. 2
x + y + z - w = 2
3x + y - z + w = 8
----------------------
4x + 2y = 10
It can be divided through by 2
2x + y = 5
3. Pick two more equations to eliminate z from.
I pick the eq. 2 and eq. 4
3x + y - z + w = 8
-x - y + z + 3w = 4
----------------------
2x + 4w = 12
It can also be divided through by 2
x + 2w = 6
3. Pick two more equations to eliminate z from.
But we must be sure this time to pick the equation we
haven't picked before.
I pick the eq. 2 and eq. 3
3x + y - z + w = 8
x - 5y + 2z + w = 1
We must multiply the top one by 2 to make the z's cancel:
6x + 2y - 2z + 2w = 16
x - 5y + 2z + w = 1
----------------------
7x - 3y + 3w = 17
Now we have this system
2x + y = 5
x + 2w = 6
7x - 3y + 3w = 17
Since y is already eliminated from the 2nd eq., we will
eliminate y from the 1st and 3rd.
2x + y = 5
7x - 3y + 3w = 17
We must multiply the top one by 3 to make the y's cancel:
6x + 3y = 15
7x - 3y + 3w = 17
------------------
13x + 3w = 32
Now we put that with the 2nd eq. and we have this system:
x + 2w = 6
13x + 3w = 32
We must multiply the top one by -13 to make the x's cancel:
-13x - 26w = -78
13x + 3w = 32
-----------------
-23w = -46
w = 2
Substitute w = 2 in
x + 2w = 6
x + 2(2) = 6
x + 4 = 6
x = 2
Substitute x = 2 in
2x + y = 5
2(2) + y = 5
4 + y = 5
y = 1
Substitute x = 2, y = 1, and w = 2 in
x + y + z - w = 2
2 + 1 + z - 2 = 2
1 + z = 2
z = 1
Solution: (x,y,z,w) = (2,1,1,2)
Edwin
