SOLUTION: Hello, I need help Please with the following problem. Solve the system: 2x + y + Z + w = 3 x - 2y - z + w = -3 x - y - 2z - w = -3 x + y

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Question 183002This question is from textbook Intermediate Algebra
: Hello,
I need help Please with the following problem.
Solve the system:
2x + y + Z + w = 3
x - 2y - z + w = -3
x - y - 2z - w = -3
x + y - z + 2w = 4

I have tried adding 1 & 2 getting 3x - y + 2w = 0
Then taking 1 & multiplying it by 2 and then adding it to 3 and getting 5x+y=w=3
Then taking 4 & mutliplying it by -2 then adding it to 1 getting 3x+2y+3w=1
Then I am stuck because I don't know what to do next. I know I have to get rid of the same 2 variables. Any help would be greatly appreciated.
Thanks,
Jen
This question is from textbook Intermediate Algebra

Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website!
2x + y + Z + w = 3
x - 2y - z + w = -3
x - y - 2z - w = -3
x + y - z + 2w = 4
-----------------------
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
----------------------
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
---------------------------
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
----------------------------
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
------------------------
Solve 3rd and 4th as a systme:
8Z + 9w = 9
3z + 7w = 7
-----------------
Multiply 1st equation by 3 and the 2nd equation by 8 to get:
24z + 27w = 27
24z + 56w = 56
-------------------
Subtract the 1st equation from the 2nd equation to get:
29w = 29
w = 1
---
Substitute to solve for "z":
3z + 56 = 56
z = 0
----------------
Substitute those values in the following equations to solve for x and for y.
x - 2y - z + w = -3
0 + y - z - 2w = 0
-------------------------
x -2y - 0 + 1 = -3
0 + y - 0 - 2 = 0
------------------------
x-2y = -4
y = 2
------------
Therefore x - 2*2 = -4
x = 0
==================
Ans: x = 0 ; y = 2 ; z = 0 ; w = 1
====================================
Cheers,
Stan H.

SOLUTION: Hello, I need help Please with the following problem. Solve the system: 2x + y + Z + w = 3 x - 2y - z + w = -3 x - y - 2z - w = -3 x + y - z + 2w = 4 I have tried add