SOLUTION: 3x+y-4z=-6 2x+3y-4z=4 x-y+2z=-4

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Question 1152439: 3x+y-4z=-6
2x+3y-4z=4
x-y+2z=-4
Found 2 solutions by Alan3354, MathLover1:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
3x+y-4z=-6
2x+3y-4z=4
x-y+ 2z=-4
------------
Multiply the 3rd eqn by 2 --> 2x - 2y + 4z = -8
----
Add that to the 1st and to the 2nd eqn.
z is eliminated.
You have 2 eqns in x & y.
--------------
3x +  y - 4z = -6
2x - 2y + 4z = -8
------------------------------ Add
5x - y = -14  ----  Eqn A
==========================
2x + 3y - 4z = 4
2x - 2y + 4z = -8
------------------------------- Add
4x + y = -4  ----   Eqn B
============================
5x - y = -14  ----  Eqn A
4x + y = -4  ----   Eqn B
-------------------------------- Add
9x = -18
x = -2
=============
4x + y = -4  ----   Eqn B
-8 + y = -4
y = 4
=====================
3x+y-4z=-6  -------- 1st eqn
-6 + 4 - 4z = -6
4z = 4
z = 1
----> (-2,4,1)

=======================================================
Mathlover used substitution.
I think that takes more work (in this case) than elimination.
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I would use a matrix because I have an Excel sheet that does those, but it doesn't teach anything, just gives the answers.

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

3x%2By-4z=-6..........1
2x%2B3y-4z=4.........2
x-y%2B2z=-4............3

start with

x-y%2B2z=-4............3...solve for+x
x=y-2z-4............3a
then,
2x%2B3y-4z=4.........2..solve for x
x=4z%2F2-3y%2F2%2B4%2F2
x=2z-3y%2F2%2B2.........2a

from 2a and 3a we have

y-2z-4=2z-3y%2F2%2B2.......solve for y
y%2B3y%2F2=2z%2B2z%2B2%2B4
5y%2F2=4z%2B6
5y=8z%2B12
y=8z%2F5%2B12%2F5................2b
go to
x=y-2z-4............3a, substitute y
x=8z%2F5%2B12%2F5-2z-4
x=8z%2F5%2B12%2F5-10z%2F5-20%2F5
x=-2z%2F5-8%2F5...........3b

go to
3x%2By-4z=-6..........1...substitute x and y, and solve for+z
3%28-2z%2F5-8%2F5%29%2B%288z%2F5%2B12%2F5%29-4z=-6
-6z%2F5-24%2F5%2B8z%2F5%2B12%2F5-20z%2F5=-6
-%2818+z%29%2F5+-+12%2F5+=+-6
6+-+12%2F5+=+%2818+z%29%2F5
%2818+z%29%2F5=30%2F2-12%2F5%0D%0A%0D%0A%7B%7B%7B%2818+%2F5%29z=18%2F5
z=1



go to
x=-2z%2F5-8%2F5...........3b.....substitute z
x=-%282%2A1%29%2F5-8%2F5...........3b
x=-2%2F5-8%2F5+
x=-10%2F5+
x=-2
go to
y=8z%2F5%2B12%2F5................2b...substitute z
y=%288%2A1%29%2F5%2B12%2F5
y=8%2F5%2B12%2F5
y=20%2F5
y=4
solutions:
x+=+-2+
y+=+4+
z+=+1