Question 152209: solve the system using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination:
w+x+y+z=5
w+2x-y-2z=-1
w-3x-3y-z=-1
2w-x+2y-z=-2
Found 2 solutions by nabla, richwmiller:
Answer by nabla(475)   (Show Source):
You can put this solution on YOUR website! 1 1 1 1 5
1 2 -1 -2 -1
1 -3 -3 -1 -1
2 -1 2 -1 -2
-R1+R2, -R1+R3
1 1 1 1 5
0 1 -2 -3 -6
0 -4 -4 -2 -6
2 -1 2 -1 -2
-2R1+R4
1 1 1 1 5
0 1 -2 -3 -6
0 -4 -4 -2 -6
0 -3 0 -3 -12
-1/3R4, -1/2R3
1 1 1 1 5
0 1 -2 -3 -6
0 2 2 1 3
0 1 0 1 4
-R2+R1, -R2+R4
1 0 3 4 11
0 1 -2 -3 -6
0 2 2 1 3
0 0 2 4 10
-2R2+R3
1 0 3 4 11
0 1 -2 -3 -6
0 0 6 7 15
0 0 2 4 10
1/2R4 <-> R3
1 0 3 4 11
0 1 -2 -3 -6
0 0 1 2 5
0 0 6 7 15
-3R3+R1,2R3+R2, -6R3+R4
1 0 0 -2 -4
0 1 0 1 4
0 0 1 2 5
0 0 0 -5 -3
-1/5R5
1 0 0 -2 -4
0 1 0 1 4
0 0 1 2 5
0 0 0 1 3/5
-2R4+R3,-R4+R2, 2R4+R1
1 0 0 0 -14/5
0 1 0 0 17/5
0 0 1 0 19/5
0 0 0 1 3/5
This gives solution set {-14/5, 17/5, 19/5, 3/5}.
Answer by richwmiller(17219)  (Show Source):
You can put this solution on YOUR website! w+x+y+z=5
w+2x-y-2z=-1
w-3x-3y-z=-1
2w-x+2y-z=-2
1,1,1,1,5
1,2,-1,-2,-1
1,-3,-3,-1,-1
2,-1,2,-1,-2
add down (-1/1) *row 1 to row 2
1,1,1,1,5
0,1,-2,-3,-6
1,-3,-3,-1,-1
2,-1,2,-1,-2
add down (-1/1) *row 1 to row 3
1,1,1,1,5
0,1,-2,-3,-6
0,-4,-4,-2,-6
2,-1,2,-1,-2
add down (-2/1) *row 1 to row 4
1,1,1,1,5
0,1,-2,-3,-6
0,-4,-4,-2,-6
0,-3,0,-3,-12
add down (4/1) *row 2 to row 3
1,1,1,1,5
0,1,-2,-3,-6
0,0,-12,-14,-30
0,-3,0,-3,-12
add down (3/1) *row 2 to row 4
1,1,1,1,5
0,1,-2,-3,-6
0,0,-12,-14,-30
0,0,-6,-12,-30
divide row 3 by -12/1
1,1,1,1,5
0,1,-2,-3,-6
0,0,1,-14/-12,-30/-12
0,0,-6,-12,-30
add down (6/1) *row 3 to row 4
1,1,1,1,5
0,1,-2,-3,-6
0,0,1,7/6,5/2
0,0,0,-5,-15
divide row 4 by -5/1
1,1,1,1,5
0,1,-2,-3,-6
0,0,1,7/6,5/2
0,0,0,1,3
This is where you would start back substitution.
We now have z=3
We continue with the matrix solution
add up (-7/6) *row 4 to row 3
1,1,1,1,5
0,1,-2,-3,-6
0,0,1,0,-1
0,0,0,1,3
now we have y= -1
add up (3/1) *row 4 to row 2
1,1,1,1,5
0,1,-2,0,3
0,0,1,0,-1
0,0,0,1,3
add up (-1/1) *row 4 to row 1
row 1 col 2
1,1,1,0,2
0,1,-2,0,3
0,0,1,0,-1
0,0,0,1,3
add up (2/1) *row 3 to row 2
1,1,1,0,2
0,1,0,0,1
0,0,1,0,-1
0,0,0,1,3
now we have z=3 y=-1 x=1
add up (-1/1) *row 3 to row 1
1,1,0,0,3
0,1,0,0,1
0,0,1,0,-1
0,0,0,1,3
add up (-1/1) *row 2 to row 1
1,0,0,0,2
0,1,0,0,1
0,0,1,0,-1
0,0,0,1,3
now we have all four solutions
w=2 x=1 y=-1 z=3
(2,1,-1,3)
w+x+y+z=5
check
2+1-1-3=5
ok
w+2x-y-2z=-1
2+2-(-1)-6=-1
5-6=-1
ok
w-3x-3y-z=-1
2-3-3(-1)-3=-1
-1+3-3=-1
ok
2w-x+2y-z=-2
2*2-1+2(-1)-3=-2
4-1-2-3=-2
3-2-3=-2
ok
(2,1,-1,3) works in all 4 equations
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