We have the equations:
4x - y + 3z = 12 ...(1)
2x + 9z = -5 ...(2)
x + 4y + 6z = -32 ...(3)
From equation (2), we can express x in terms of z:
2x = -9z - 5
x = (-9z - 5) / 2 ...(4)
Substitute equation (4) into equations (1) and (3):
From equation (1):
4((-9z - 5) / 2) - y + 3z = 12
-18z - 10 - y + 3z = 12
-15z - y = 22 ...(5)
From equation (3):
((-9z - 5) / 2) + 4y + 6z = -32
-9z - 5 + 8y + 12z = -64
3z + 8y = -59 ...(6)
Now we have a system of two equations in terms of y and z.
To solve the system, let's multiply equation (5) by 3 and equation (6) by 15 to eliminate z:
-45z - 3y = 66 ...(7)
45z + 120y = -885 ...(8)
Add equations (7) and (8):
-3y + 120y = 66 - 885
117y = -819
y = -7
Substitute y = -7 into equation (7):
-45z - 3(-7) = 66
-45z + 21 = 66
-45z = 45
z = -1
Substitute y = -7 and z = -1 into equation (4):
x = (-9(-1) - 5) / 2
x = (9 - 5) / 2
x = 2
Therefore, the solution to the system of equations is:
x = 2
y = -7
z = -1