3x + 2y - z = 8
-3x + 4y + 5z = -14
x - 3y + 4z = -14
add the 2nd equation to the first
0x + 6y + 4z = -6
-3x + 4y + 5z = -14
x - 3y + 4z = -14
multiply the bottom by 3 on both sides
0x + 6y + 4z = -6
-3x + 4y + 5z = -14
3x - 9y + 12z = -42
add the middle to the bottom
0x + 6y + 4z = -6
-3x + 4y + 5z = -14
0x - 5y + 17z = -56
multiply the top by 5 and the bottom by 6
0x + 30y + 20z = -30
-3x + 4y + 5z = -14
0x - 30y + 102z = -336
add the bottom to the top
0x + 0y + 122z = -366
-3x + 4y + 5z = -14
0x - 30y + 102z = -336
divide the top by 122
0x + 0y + z = -3
-3x + 4y + 5z = -14
0x - 30y + 102z = -336
multiply the top equation by -102
0x + 0y - 102z = 306
-3x + 4y + 5z = -14
0x - 30y + 102z = -336
add the top to the bottom
0x + 0y - 102z = 306
-3x + 4y + 5z = -14
0x - 30y + 0z = -30
divide the bottom by -30 and the top by -102
0x + 0y + z = -3
-3x + 4y + 5z = -14
0x + y + 0z = 1
multiply the top by -5 and the bottom by -4
0x + 0y - 5z = 15
-3x + 4y + 5z = -14
0x - 4y + 0z = -4
add the top and bottom to the middle
0x + 0y - 5z = 15
-3x + 0y + 0z = -3
0x - 4y + 0z = -4
divide the top by -5, the middle by -3, and the bottom by -4
0x + 0y + z = -3
x + 0y + 0z = 1
0x + y + 0z = 1
Answer: x = 1, y = 1, z = -3