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 find the complete solution to the system of equations, we need to perform a series of row operations to reduce the system into row-echelon form (also known as reduced row-echelon form).

Let's start by writing the system of equations in augmented matrix form:

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

To eliminate the first coefficient below the main diagonal (in this case, the (2,1) entry), we'll multiply the first row by -1/4 and add it to the second row. We'll also multiply the first row by -5/4 and add it to the third row. This is the first step of Gaussian elimination:

```plaintext
[ 4 -1 3 | 12 ]
[ 0 17/4 3/4 | -35 ]
[ 0 23/4 6/4 | -28 ]
```

Next, we'll divide the second row by 17/4 to make the coefficient of y in the second equation equal to 1. Similarly, we'll divide the third row by 23/4 to make the coefficient of y in the third equation equal to 1:

```plaintext
[ 4 -1 3 | 12 ]
[ 0 1 3/17 | -35/17 ]
[ 0 1 6/23 | -28/23 ]
```

To eliminate the coefficient below the main diagonal in the third row, we'll multiply the second row by -1 and add it to the third row:

```plaintext
[ 4 -1 3 | 12 ]
[ 0 1 3/17 | -35/17 ]
[ 0 0 69/391 | -231/391 ]
```

Now, we'll divide the third row by 69/391 to make the coefficient of z in the third equation equal to 1:

```plaintext
[ 4 -1 3 | 12 ]
[ 0 1 3/17 | -35/17 ]
[ 0 0 1 | -231/69 ]
```

Finally, let's solve for the variables using back substitution. Starting from the last row, we have:

z = -231/69

Substituting this value of z in the second equation, we get:

y + (3/17) * (-231/69) = -35/17
y - (1/17) * (-231/23) = -35/17
y + 11 = -35/17
y = -35/17 - 11
y = -52/17

Substituting the values of y and z in the first equation, we have:

4x - (-52/17) + 3(-231/69) = 12
4x + 52/17 - 693/69 = 12
4x + 52/17 - 10 = 12
4x = 10 - 52/17
4x = (170 - 52)/17
4x = 118/17
x = (118/17) * (1/4)
x = 118/68
x = 59/34

Therefore, the complete solution to the system of equations is:
x = 59/34, y = -52/17, z = -231/69.