SOLUTION: x+2y+3z=1,3x-y+z=1,4x+2y+z=3

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Question 983847: x+2y+3z=1,3x-y+z=1,4x+2y+z=3
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
x%2B2y%2B3z=1..........eq.1
3x-y%2Bz=1..........eq.2
4x%2B2y%2Bz=3..........eq.3
--------------------------------------
start with
x%2B2y%2B3z=1..........eq.1.....both sides multiply by 3
3x-y%2Bz=1..........eq.2
-------------------------------

3x%2B6y%2B9z=3..........eq.1
3x-y%2Bz=1..........eq.2
---------------------------------subtract eq.2 from eq.1
3x%2B6y%2B9z-%283x-y%2Bz%29=3-1
cross%283x%29%2B6y%2B9z-cross%283x%29%2By-z=2
7y%2B8z=2.......solve for y
7y=2-8z
y=%282-8z%29%2F7%7D%7D.......eq.1a%0D%0A%0D%0Anow%2C+go+to%0D%0A%0D%0A%7B%7B%7Bx%2B2y%2B3z=1..........eq.1...both sides multiply by 4
4x%2B2y%2Bz=3..........eq.3
-------------------------------------
4x%2B8y%2B12z=4..........eq.1
4x%2B2y%2Bz=3..........eq.3
-------------------------------------subtract eq.3 from eq.1
4x%2B8y%2B12z-%284x%2B2y%2Bz%29=4-3
cross%284x%29%2B8y%2B12z-cross%284x%29-2y-z=1
6y%2B11z=1......solve for y
6y=1-11z
y=%281-11z%29%2F6.......eq.2a
since eq.1a and eq.2a have same left sides, then right sides are equal too
%282-8z%29%2F7=%281-11z%29%2F6...........solve for z
%282-8z%296=7%281-11z%29
12-48z=7-77z
77z-48z=7-12
29z=-5
highlight%28z=-5%2F29%29
go to y=%281-11z%29%2F6.......eq.2a, substitute -5%2F29 for z and find y
y=%281-11%28-5%2F29%29%29%2F6
y=%281%2B55%2F29%29%2F6
y=%2829%2F29%2B55%2F29%29%2F6
y=%28cross%2884%2914%2F29%29%2Fcross%286%291
highlight%28y=14%2F29%29
go to x%2B2y%2B3z=1..........eq.1 substitute -5%2F29 for z,14%2F29 for y and find x
x%2B2%2814%2F29%29%2B3%28-5%2F29%29=1..........eq.1
x%2B28%2F29-15%2F29=1
x%2B13%2F29=1
x=1-313%2F29
x=29%2F29-13%2F29
highlight%28x=16%2F29%29

so, your solutions are:
highlight%28x=16%2F29%29
highlight%28y=14%2F29%29
highlight%28z=-5%2F29%29