Write the augmented matrix, and then solve the system, using Gauss Jordan elimination on the

augmented matrix.
x + 2y - z = 4
-2x + y - 4z = -6
4x - 3y + 2z = -10

enter your matrix of coefficients here and watch the details unfold:

http://www.gregthatcher.com/Mathematics/GaussJordan.aspx

To solve the system using Gauss Jordan elimination, we first need to create an augmented matrix using the given equations. The augmented matrix is formed by taking the coefficients of the variables and the constants on the right-hand side of each equation. The augmented matrix for the system is as follows:

```
1 2 -1 | 4
-2 1 -4 | -6
4 -3 2 | -10
```

Now, we will perform row operations on the augmented matrix to transform it into row-echelon form and then into reduced row-echelon form.

Step 1: Swap rows R1 and R2

```
-2 1 -4 | -6
1 2 -1 | 4
4 -3 2 | -10
```

Step 2: Multiply R1 by -1/2 and add it to R2, Multiply R1 by -2 and add it to R3

```
-2 1 -4 | -6
0 4 -3 | 7
0 -1 6 | -2
```

Step 3: Multiply R2 by 1/4 and add it to R1, Multiply R2 by 1/4 and add it to R3

```
-2 0 -5/2 | -7/2
0 4 -3 | 7
0 0 3/2 | 1/2
```

Step 4: Multiply R3 by 2/3

```
-2 0 -5/2 | -7/2
0 4 -3 | 7
0 0 1 | 1/3
```

Step 5: Multiply R2 by 5/2 and add it to R1, Multiply R2 by 3 and add it to R3

```
-2 0 0 | 6
0 4 0 | 31/2
0 0 1 | 1/3
```

Step 6: Multiply R1 by -1/2

```
1 0 0 | -3
0 4 0 | 31/2
0 0 1 | 1/3
```

The augmented matrix is now in reduced row-echelon form. From the last column of the augmented matrix, we can determine the values of the variables:

x = -3
y = 31/8
z = 1/3

Therefore, the solution to the system of equations is x = -3, y = 31/8, and z = 1/3.

To write the augmented matrix, we need to represent the system of equations in a matrix form. Each equation will be a row in the matrix, and the coefficients of the variables will be the entries.

The system of equations:
x + 2y - z = 4 --(1)
-2x + y - 4z = -6 --(2)
4x - 3y + 2z = -10 --(3)

The augmented matrix will have the coefficients of the variables as its entries, with an additional column for the constants on the right-side of the equations. The matrix will look like this:

| 1 2 -1 | 4 |
| -2 1 -4 | -6 |
| 4 -3 2 | -10|

Now, let's solve this system of equations using Gauss Jordan elimination on the augmented matrix.

Step 1: Let's start with the first row and eliminate the x-coefficient in the second and third rows. To do this, we need to multiply the first row by 2 and add it to the second row. Similarly, multiply the first row by -4 and add it to the third row.

| 1 2 -1 | 4 |
| 0 5 -6 | 2 |
| 0 -11 6 | 6 |

Step 2: Now, we will eliminate the y-coefficient in the first and third rows. Multiply the second row by -2 and add it to the first row. Multiply the second row by 11 and add it to the third row.

| 1 0 11 | 0 |
| 0 5 -6 | 2 |
| 0 0 0 | 28 |

Step 3: Finally, we will solve for the variables by performing row operations. Divide the first row by 1, the second row by 5, and the third row by 28.

| 1 0 11 | 0 |
| 0 1 -6/5 | 2/5 |
| 0 0 0 | 1 |

The last row of the augmented matrix is inconsistent because it has all zeros except for the constant term 1. This tells us that the system of equations is inconsistent and does not have a solution.

Therefore, the system of equations is inconsistent, and there is no solution.