2x + 4y − 3z = −7 ---> first equation
3x + y + 4z = −12 ---> second equation
x + 3y + 4z = 4 ---> third equation
We try to eliminate z and make two equations
Multiply first equation by 4 and second equation by 3. then add both equations
8x + 16y -12z = -28
9x + 3y +12z = -36
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17x + 19y = -64 ---------------> equation 4
multiply third equation by -1 and add second and third equation
3x + y + 4z = −12
-x - 3y - 4z = -4
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2x -2y = -16 ---------------> equation 5
Multiply equation 4 by 2 and equation by 19
34x + 38y = -128
38x - 38y = -304
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72x = -432
Divide by 72 on both sides
x = -6
use equation 2x -2y = -16 ---------------> equation 5
2(-6) - 2y = -16
-12 -2y = -16
-2y = -4
So y = 2
Now plug in 2 for y and -6 for x
Using any one of the three equations
x + 3y + 4z = 4
-6 + 3(2) +4z = 4
4z = 4 (divide by 4 on both sides)
So z=1
So x= -6, y = 2 and z= 1