Answer:
x = -262/13, y = -5/13, z = 89/13
Explanation:
Solve the following system:
{3 z + 4 y = 19 | (equation 1)
z - x = 27 | (equation 2)
-3 y + x = -19 | (equation 3)
Swap equation 1 with equation 2:
{-x + 0 y + z = 27 | (equation 1)
0 x + 4 y + 3 z = 19 | (equation 2)
x - 3 y + 0 z = -19 | (equation 3)
Add equation 1 to equation 3:
{-x + 0 y + z = 27 | (equation 1)
0 x + 4 y + 3 z = 19 | (equation 2)
0 x - 3 y + z = 8 | (equation 3)
Add 3/4 × (equation 2) to equation 3:
{-x + 0 y + z = 27 | (equation 1)
0 x + 4 y + 3 z = 19 | (equation 2)
0 x + 0 y + (13 z)/4 = 89/4 | (equation 3)
Multiply equation 3 by 4:
{-x + 0 y + z = 27 | (equation 1)
0 x + 4 y + 3 z = 19 | (equation 2)
0 x + 0 y + 13 z = 89 | (equation 3)
Divide equation 3 by 13:
{-x + 0 y + z = 27 | (equation 1)
0 x + 4 y + 3 z = 19 | (equation 2)
0 x + 0 y + z = 89/13 | (equation 3)
Subtract 3 × (equation 3) from equation 2:
{-x + 0 y + z = 27 | (equation 1)
0 x + 4 y + 0 z = -20/13 | (equation 2)
0 x + 0 y + z = 89/13 | (equation 3)
Divide equation 2 by 4:
{-x + 0 y + z = 27 | (equation 1)
0 x + y + 0 z = -5/13 | (equation 2)
0 x + 0 y + z = 89/13 | (equation 3)
Subtract equation 3 from equation 1:
{-x + 0 y + 0 z = 262/13 | (equation 1)
0 x + y + 0 z = -5/13 | (equation 2)
0 x + 0 y + z = 89/13 | (equation 3)
Multiply equation 1 by -1:
{x + 0 y + 0 z = -262/13 | (equation 1)
0 x + y + 0 z = -5/13 | (equation 2)
0 x + 0 y + z = 89/13 | (equation 3)
Collect results:
Answer: {x = -262/13, y = -5/13, z = 89/13