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