Solution for:

-x + 3y - 2z = 19

2x + y - z = 5

-3x - y + 2z = - 7

x = - 1, y = 4, z = - 3

Step-by-step explanation:

Solve the following system:

{-x + 3 y - 2 z = 19 | (equation 1)

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

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

Swap equation 1 with equation 3:

{ - (3 x) - y + 2 z = - 7 | (equation 1)

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

-x + 3 y - 2 z = 19 | (equation 3)

Add 2/3 * (equation 1) to equation 2:

{ - (3 x) - y + 2 z = - 7 | (equation 1)

0 x+y/3 + z/3 = 1/3 | (equation 2)

-x + 3 y - 2 z = 19 | (equation 3)

Multiply equation 2 by 3:

{ - (3 x) - y + 2 z = - 7 | (equation 1)

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

-x + 3 y - 2 z = 19 | (equation 3)

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

{ - (3 x) - y + 2 z = - 7 | (equation 1)

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

0 x + (10 y) / 3 - (8 z) / 3 = 64/3 | (equation 3)

Multiply equation 3 by 3/2:

{ - (3 x) - y + 2 z = - 7 | (equation 1)

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

0 x+5 y - 4 z = 32 | (equation 3)

Swap equation 2 with equation 3:

{ - (3 x) - y + 2 z = - 7 | (equation 1)

0 x+5 y - 4 z = 32 | (equation 2)

0 x+y + z = 1 | (equation 3)

Subtract 1/5 * (equation 2) from equation 3:

{ - (3 x) - y + 2 z = - 7 | (equation 1)

0 x+5 y - 4 z = 32 | (equation 2)

0 x+0 y + (9 z) / 5 = (-27) / 5 | (equation 3)

Multiply equation 3 by 5/9:

{ - (3 x) - y + 2 z = - 7 | (equation 1)

0 x+5 y - 4 z = 32 | (equation 2)

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

Add 4 * (equation 3) to equation 2:

{ - (3 x) - y + 2 z = - 7 | (equation 1)

0 x+5 y+0 z = 20 | (equation 2)

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

Divide equation 2 by 5:

{ - (3 x) - y + 2 z = - 7 | (equation 1)

0 x+y+0 z = 4 | (equation 2)

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

Add equation 2 to equation 1:

{ - (3 x) + 0 y+2 z = - 3 | (equation 1)

0 x+y+0 z = 4 | (equation 2)

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

Subtract 2 * (equation 3) from equation 1:

{ - (3 x) + 0 y+0 z = 3 | (equation 1)

0 x+y+0 z = 4 | (equation 2)

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

Divide equation 1 by - 3:

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

0 x+y+0 z = 4 | (equation 2)

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

Collect results:

Answer: {x = - 1, y = 4, z = - 3