X-6y + 4z=-12

x+y-4z=12

2x + 2y + 5z = -15

Systems of equations with three variables and three equations

x = 0, y = 0, z = - 3

Step-by-step explanation:

Solve the following system:

{x - 6 y + 4 z = - 12 | (equation 1)

x + y - 4 z = 12 | (equation 2)

2 x + 2 y + 5 z = - 15 | (equation 3)

Swap equation 1 with equation 3:

{2 x + 2 y + 5 z = - 15 | (equation 1)

x + y - 4 z = 12 | (equation 2)

x - 6 y + 4 z = - 12 | (equation 3)

Subtract 1/2 * (equation 1) from equation 2:

{2 x + 2 y + 5 z = - 15 | (equation 1)

0 x+0 y - (13 z) / 2 = 39/2 | (equation 2)

x - 6 y + 4 z = - 12 | (equation 3)

Multiply equation 2 by 2/13:

{2 x + 2 y + 5 z = - 15 | (equation 1)

0 x+0 y - z = 3 | (equation 2)

x - 6 y + 4 z = - 12 | (equation 3)

Subtract 1/2 * (equation 1) from equation 3:

{2 x + 2 y + 5 z = - 15 | (equation 1)

0 x+0 y - z = 3 | (equation 2)

0 x - 7 y + (3 z) / 2 = - 9/2 | (equation 3)

Multiply equation 3 by 2:

{2 x + 2 y + 5 z = - 15 | (equation 1)

0 x+0 y - z = 3 | (equation 2)

0 x - 14 y + 3 z = - 9 | (equation 3)

Swap equation 2 with equation 3:

{2 x + 2 y + 5 z = - 15 | (equation 1)

0 x - 14 y + 3 z = - 9 | (equation 2)

0 x+0 y - z = 3 | (equation 3)

Multiply equation 3 by - 1:

{2 x + 2 y + 5 z = - 15 | (equation 1)

0 x - 14 y + 3 z = - 9 | (equation 2)

0 x+0 y+z = - 3 | (equation 3)

Subtract 3 * (equation 3) from equation 2:

{2 x + 2 y + 5 z = - 15 | (equation 1)

0 x - 14 y+0 z = 0 | (equation 2)

0 x+0 y+z = - 3 | (equation 3)

Divide equation 2 by - 14:

{2 x + 2 y + 5 z = - 15 | (equation 1)

0 x+y+0 z = 0 | (equation 2)

0 x+0 y+z = - 3 | (equation 3)

Subtract 2 * (equation 2) from equation 1:

{2 x + 0 y+5 z = - 15 | (equation 1)

0 x+y+0 z = 0 | (equation 2)

0 x+0 y+z = - 3 | (equation 3)

Subtract 5 * (equation 3) from equation 1:

{2 x+0 y+0 z = 0 | (equation 1)

0 x+y+0 z = 0 | (equation 2)

0 x+0 y+z = - 3 | (equation 3)

Divide equation 1 by 2:

{x+0 y+0 z = 0 | (equation 1)

0 x+y+0 z = 0 | (equation 2)

0 x+0 y+z = - 3 | (equation 3)

Collect results:

Answer: {x = 0, y = 0, z = - 3