SOLUTION: Please help without using the matrix system! thank you x+y+z-w=5 2x+y-z+w=6 x-5y+3z+w=9 -x-y+z+4w=7

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Question 907725: Please help without using the matrix system! thank you
x+y+z-w=5
2x+y-z+w=6
x-5y+3z+w=9
-x-y+z+4w=7

Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
OK so use row operations and elimination to reduce your variables.
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R%5B1%5D:x%2By%2Bz-w=5
R%5B2%5D:2x%2By-z%2Bw=6
R%5B3%5D:x-5y%2B3z%2Bw=9
R%5B4%5D:-x-y%2Bz%2B4w=7
Let's get rid of w first.
R%5B5%5D=R%5B1%5D%2BR%5B2%5D: x%2By%2Bz-w%2B2x%2By-z%2Bw=5%2B6
R%5B5%5D=R%5B1%5D%2BR%5B2%5D: 3x%2B2y=11
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R%5B6%5D=R%5B1%5D%2BR%5B3%5D: x%2By%2Bz-w%2Bx-5y%2B3z%2Bw=5%2B9
R%5B6%5D=R%5B1%5D%2BR%5B3%5D: 2x-4y%2B4z=14
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R%5B7%5D=4%2AR%5B1%5D%2BR%5B4%5D: 4x%2B4y%2B4z-4w-x-y%2Bz%2B4w=20%2B7
R%5B7%5D=4%2AR%5B1%5D%2BR%5B4%5D: 3x%2B3y%2B5z=27
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OK, R%5B5%5D already has z eliminated, so let's pair up R%5B6%5D and R%5B7%5D to do the same.
R%5B8%5D=5%2AR%5B6%5D-4%2AR%5B7%5D:10x-20y%2B20z-12x-12y-20z=70-108
R%5B8%5D=5%2AR%5B6%5D-4%2AR%5B7%5D:-2x-32y=-38
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Finally, multiply R%5B5%5D by 16 and add to R%5B8%5D to eliminate y.
R%5B9%5D=16%2AR5%2BR8 : 48x%2B32y-2x-32y=176-38
R%5B9%5D=16%2AR5%2BR8 : 46x=148
So then,
x=148%2F46
x=3
Then work backwards,
R%5B5%5D: 3%283%29%2B2y=11
2y=2
y=1
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R%5B6%5D: 2%283%29-4%281%29%2B4z=14
6-4%2B4z=14
4z=12
z=3
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R%5B1%5D:3%2B1%2B3-w=5
w=2