SOLUTION: Find all solutions of the system x+y+z+w=4 x-2y-z-w=3 2x-y+z-w=2 x-y+2z-2w=-7 I need to use Gaussian elimination but don't know how with 4 variables. Please help asap!

Question 881485: Find all solutions of the system
x+y+z+w=4
x-2y-z-w=3
2x-y+z-w=2
x-y+2z-2w=-7
I need to use Gaussian elimination but don't know how with 4 variables. Please help asap!

Answer by richwmiller(17219) (Show Source): You can put this solution on YOUR website!
1,1,1,1,4
1,-2,-1,-1,3
2,-1,1,-1,2
1,-1,2,-2,-7

add down (-1) *row 1 to row 2
1,1,1,1,4
0,-3,-2,-2,-1
2,-1,1,-1,2
1,-1,2,-2,-7
add down (-2) *row 1 to row 3
1,1,1,1,4
0,-3,-2,-2,-1
0,-3,-1,-3,-6
1,-1,2,-2,-7
add down (-1) *row 1 to row 4
1,1,1,1,4
0,-3,-2,-2,-1
0,-3,-1,-3,-6
0,-2,1,-3,-11
divide row 2 by -3
1,1,1,1,4
0,1,-2/-3,-2/-3,-1/-3
0,-3,-1,-3,-6
0,-2,1,-3,-11
add down (3) *row 2 to row 3
1,1,1,1,4
0,1,2/3,2/3,1/3
0,0,1,-1,-5
0,-2,1,-3,-11
add down (2) *row 2 to row 4
1,1,1,1,4
0,1,2/3,2/3,1/3
0,0,1,-1,-5
0,0,7/3,-5/3,-31/3

add down (-7/3) *row 3 to row 4
1,1,1,1,4
0,1,2/3,2/3,1/3
0,0,1,-1,-5
0,0,0,6/9,12/9
divide row 4 by 2/3
1,1,1,1,4
0,1,2/3,2/3,1/3
0,0,1,-1,-5
0,0,0,1,2
We now have the value for the last variable.
We will work our way up and get the other solutions.
add up (1) *row 4 to row 3
1,1,1,1,4
0,1,2/3,2/3,1/3
0,0,1,0,-3
0,0,0,1,2
add up (-2/3) *row 4 to row 2
1,1,1,1,4
0,1,6/9,0,-1
0,0,1,0,-3
0,0,0,1,2
add up (-1) *row 4 to row 1
1,1,1,0,2
0,1,2/3,0,-1
0,0,1,0,-3
0,0,0,1,2
add up (-2/3) *row 3 to row 2
1,1,1,0,2
0,1,0,0,1
0,0,1,0,-3
0,0,0,1,2
add up (-1) *row 3 to row 1
1,1,0,0,5
0,1,0,0,1
0,0,1,0,-3
0,0,0,1,2
add up (-1) *row 2 to row 1
1,0,0,0,4
0,1,0,0,1
0,0,1,0,-3
0,0,0,1,2
final
1,0,0,0,4
0,1,0,0,1
0,0,1,0,-3
0,0,0,1,2
"4","1","-3","2"
(4,1,-3,2)
x,y,z,w
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