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

4x-y+3z = 12
x+4y+6z =-32
5x+3y+9z = 20

To use Gaussian elimination to solve the system of equations, we'll start by writing the augmented matrix for the system:

[ 4 -1 3 | 12 ]
[ 1 4 6 | -32 ]
[ 5 3 9 | 20 ]

Step 1: Rearrange the rows to put the row with the largest element at the top.

[ 5 3 9 | 20 ]
[ 4 -1 3 | 12 ]
[ 1 4 6 | -32 ]

Step 2: Use row operations to eliminate the coefficients below the leading 1s in the first column and make the other elements below them zero.

- (4/5) * R1 + R2 -> R2
- (1/5) * R1 + R3 -> R3

[ 5 3 9 | 20 ]
[ 0 -7/5 -9/5 | -32/5 ]
[ 0 17/5 -3/5 | -36/5 ]

Step 3: Use row operations to eliminate the coefficients below the leading 1 in the second row and make the other elements below it zero.

(17/5) * R2 + R3 -> R3

[ 5 3 9 | 20 ]
[ 0 -7/5 -9/5 | -32/5 ]
[ 0 0 0 | -44/5 ]

Step 4: Check for inconsistency
Since there is a row of zeros on the left side and a non-zero number (0) on the right side, the system is inconsistent. This means that there is no solution that satisfies all three equations simultaneously.

Therefore, the system of equations has no solution.