Solve each system by elimination

-x-5y-5z=2

4x-5y+4z=19

x+5y-z=-20

x = - 2, y = - 3, z = 3

Step-by-step explanation:

Solve the following system:

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

4 x - 5 y + 4 z = 19 | (equation 2)

x + 5 y - z = - 20 | (equation 3)

Swap equation 1 with equation 2:

{4 x - 5 y + 4 z = 19 | (equation 1)

-x - 5 y - 5 z = 2 | (equation 2)

x + 5 y - z = - 20 | (equation 3)

Add 1/4 * (equation 1) to equation 2:

{4 x - 5 y + 4 z = 19 | (equation 1)

0 x - (25 y) / 4 - 4 z = 27/4 | (equation 2)

x + 5 y - z = - 20 | (equation 3)

Multiply equation 2 by 4:

{4 x - 5 y + 4 z = 19 | (equation 1)

0 x - 25 y - 16 z = 27 | (equation 2)

x + 5 y - z = - 20 | (equation 3)

Subtract 1/4 * (equation 1) from equation 3:

{4 x - 5 y + 4 z = 19 | (equation 1)

0 x - 25 y - 16 z = 27 | (equation 2)

0 x + (25 y) / 4 - 2 z = - 99/4 | (equation 3)

Multiply equation 3 by 4:

{4 x - 5 y + 4 z = 19 | (equation 1)

0 x - 25 y - 16 z = 27 | (equation 2)

0 x+25 y - 8 z = - 99 | (equation 3)

Add equation 2 to equation 3:

{4 x - 5 y + 4 z = 19 | (equation 1)

0 x - 25 y - 16 z = 27 | (equation 2)

0 x+0 y - 24 z = - 72 | (equation 3)

Divide equation 3 by - 24:

{4 x - 5 y + 4 z = 19 | (equation 1)

0 x - 25 y - 16 z = 27 | (equation 2)

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

Add 16 * (equation 3) to equation 2:

{4 x - 5 y + 4 z = 19 | (equation 1)

0 x - 25 y+0 z = 75 | (equation 2)

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

Divide equation 2 by - 25:

{4 x - 5 y + 4 z = 19 | (equation 1)

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

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

Add 5 * (equation 2) to equation 1:

{4 x + 0 y+4 z = 4 | (equation 1)

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

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

Subtract 4 * (equation 3) from equation 1:

{4 x+0 y+0 z = - 8 | (equation 1)

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

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

Divide equation 1 by 4:

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

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

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

Collect results:

Answer: {x = - 2, y = - 3, z = 3