Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. Show your work.


x+4y+6z =-32
5x+3y+9z = 20

When you want to solve system with three unknowns you must write three equations, not two.

Yeah But I already did the first one , sorry. The first one is 4x-y+3z = 12

To solve the system of equations using Gaussian elimination, we need to rewrite the system in augmented matrix form.

The augmented matrix form of the system is:
[1 4 6 | -32]
[5 3 9 | 20]

Step 1: Pivot the first row by dividing it by the coefficient of x (i.e., 1).
[1 4 6 | -32]
[5 3 9 | 20]

Step 2: Use row operations to eliminate the coefficient of x in the second row.
R2 = R2 - 5R1
[1 4 6 | -32]
[0 -17 -21 | 180]

Step 3: Pivot the second row by dividing it by the coefficient of y (i.e., -17).
[1 4 6 | -32]
[0 1 1.24 | -10.588]

Step 4: Use row operations to eliminate the coefficient of y in the first row.
R1 = R1 - 4R2
[1 0 1.04 | 7.336]
[0 1 1.24 | -10.588]

Step 5: Pivot the third column by dividing it by the coefficient of z (i.e., 1.04).
[1 0 1.04 | 7.336]
[0 1 1.24 | -10.588]

Step 6: Use row operations to eliminate the coefficient of z in the first row.
R1 = R1 - 1.04R3
[1 0 0 | 0.032]
[0 1 1.24 | -10.588]

Therefore, the complete solution to the given system of equations is:
x = 0.032
y = -10.588 - 1.24z
z = z

Hence, the solution can be written as:
x = 0.032
y = -10.588 - 1.24z
z = z, where z is any real number.