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:

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