Solve x + y + z = 12 2x – y – z = -6 x + 3y + 5z = 44 (4, 3, 5) no solutions (2, 4, -6) (2, 4, 6)

The answer is (2,4,6)
Proof:

Solve the following system:{x + y + z = 12 | (equation 1){2 x - y - z = -6 | (equation 2){x + 3 y + 5 z = 44 | (equation 3)
Swap equation 1 with equation 2:{2 x - y - z = -6 | (equation 1){x + y + z = 12 | (equation 2){x + 3 y + 5 z = 44 | (equation 3)
Subtract 1/2 × (equation 1) from equation 2:{2 x - y - z = -6 | (equation 1){0 x+(3 y)/2 + (3 z)/2 = 15 | (equation 2){x + 3 y + 5 z = 44 | (equation 3)
Multiply equation 2 by 2/3:{2 x - y - z = -6 | (equation 1){0 x+y + z = 10 | (equation 2){x + 3 y + 5 z = 44 | (equation 3)
Subtract 1/2 × (equation 1) from equation 3:{2 x - y - z = -6 | (equation 1){0 x+y + z = 10 | (equation 2){0 x+(7 y)/2 + (11 z)/2 = 47 | (equation 3)
Multiply equation 3 by 2:{2 x - y - z = -6 | (equation 1){0 x+y + z = 10 | (equation 2)v0 x+7 y + 11 z = 94 | (equation 3)
Swap equation 2 with equation 3:{2 x - y - z = -6 | (equation 1){0 x+7 y + 11 z = 94 | (equation 2){0 x+y + z = 10 | (equation 3)
Subtract 1/7 × (equation 2) from equation 3:{2 x - y - z = -6 | (equation 1){0 x+7 y + 11 z = 94 | (equation 2){0 x+0 y - (4 z)/7 = (-24)/7 | (equation 3)Multiply equation 3 by -7/4:{2 x - y - z = -6 | (equation 1){0 x+7 y + 11 z = 94 | (equation 2){0 x+0 y+z = 6 | (equation 3)
Subtract 11 × (equation 3) from equation 2:{2 x - y - z = -6 | (equation 1){0 x+7 y+0 z = 28 | (equation 2){0 x+0 y+z = 6 | (equation 3)
Divide equation 2 by 7:{2 x - y - z = -6 | (equation 1){0 x+y+0 z = 4 | (equation 2){0 x+0 y+z = 6 | (equation 3)
Add equation 2 to equation 1:{2 x + 0 y - z = -2 | (equation 1){0 x+y+0 z = 4 | (equation 2){0 x+0 y+z = 6 | (equation 3)Add equation 3 to equation 1:{2 x+0 y+0 z = 4 | (equation 1){0 x+y+0 z = 4 | (equation 2){0 x+0 y+z = 6 | (equation 3)
Divide equation 1 by 2:{x+0 y+0 z = 2 | (equation 1){0 x+y+0 z = 4 | (equation 2){0 x+0 y+z = 6 | (equation 3)
Collect results:

{x = 2, y = 4 , z = 6