Question 183002
2x + y + Z + w = 3
x - 2y - z + w = -3
x - y - 2z - w = -3
x + y - z + 2w = 4 
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Interchange 1st and 2nd to get:
x - 2y - z + w = -3
2x + y + Z + w = 3
x - y - 2z - w = -3
x + y - z + 2w = 4 
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Subtract two times the 1st equation from the 2nd; subtract the 1st equation 
from the 3rd; subtract the 1st equation from the 4th equation to get:
x - 2y - z + w = -3
0 + 5y + 3Z - w = 9
0 + y -  z - 2w = 0
0 + 3y + 0 +  w = 7 
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Interchange the 2nd equation and the 3rd equation to get:
x - 2y - z + w = -3
0 + y -  z - 2w = 0
0 + 5y + 3Z - w = 9
0 + 3y + 0 +  w = 7 
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Subtract 5 times the 2nd equation from the 3rd ; subtract 3 times the 2nd equation from the 4th to get:

x - 2y - z + w = -3
0 + y -  z - 2w = 0
0 + 0 + 8Z + 9w = 9
0 + 0 + +3z + 7w = 7 
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Solve 3rd and 4th as a systme:
8Z + 9w = 9
3z + 7w = 7
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Multiply 1st equation by 3 and the 2nd equation by 8 to get:
24z + 27w = 27
24z + 56w = 56
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Subtract the 1st equation from the 2nd equation to get:
29w = 29
w = 1
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Substitute to solve for "z":
3z + 56 = 56
z = 0
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Substitute those values in the following equations to solve for x and for y.
x - 2y - z + w = -3
0 + y -  z - 2w = 0
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x -2y - 0 + 1 = -3
0 + y - 0 - 2 = 0
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x-2y = -4
y = 2
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Therefore x - 2*2 = -4
x = 0
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Ans: x = 0 ; y = 2 ; z = 0 ; w = 1
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Cheers,
Stan H.