write and row reduce the augmented matrix to find the general solution.
x + y - z = 7
2x - y - 5z = 2
-5x + 4y + 14z = 1
3x - y - 7z = 5
To find the general solution, we need to row reduce the augmented matrix formed by the coefficients of the variables.
The augmented matrix corresponding to the given system of equations is:
| 1 1 -1 | 7 |
| 2 -1 -5 | 2 |
| -5 4 14 | 1 |
| 3 -1 -7 | 5 |
We will perform row operations to bring the augmented matrix into row-echelon form (also known as reduced row-echelon form) using the Gauss-Jordan elimination method.
Step 1: Start with the first row. Divide it by the leading coefficient (1 in this case) to make it equal to 1. The first row becomes:
| 1 1 -1 | 7 |
Step 2: Use row operations to make the entries below the leading 1 in the first column equal to 0. We achieve this by multiplying the first row by -2 (to make the second entry -2) and adding it to the second row. Similarly, multiply the first row by 5 and add it to the third row. Finally, multiply the first row by -3 and add it to the fourth row. The augmented matrix after this operation is:
| 1 1 -1 | 7 |
| 0 -3 -3 | -12 |
| 0 9 19 | 36 |
| 0 -4 -4 | -16 |
Step 3: Continue with the second row. Divide it by the leading coefficient (-3 in this case) to make it equal to 1. The second row becomes:
| 1 1 -1 | 7 |
| 0 1 1 | 4 |
| 0 9 19 | 36 |
| 0 -4 -4 | -16 |
Step 4: Use row operations to make the entries below the leading 1 in the second column equal to 0. We achieve this by multiplying the second row by -1 (to make the third entry -1) and adding it to the third row. Similarly, multiply the second row by 4 and add it to the fourth row. The augmented matrix after this operation is:
| 1 1 -1 | 7 |
| 0 1 1 | 4 |
| 0 0 10 | 0 |
| 0 0 0 | 0 |
Step 5: Continue with the third row. Divide it by the leading coefficient (10 in this case) to make it equal to 1. The third row becomes:
| 1 1 -1 | 7 |
| 0 1 1 | 4 |
| 0 0 1 | 0 |
| 0 0 0 | 0 |
Step 6: Use row operations to make the entries above the leading 1 in the third column equal to 0. We achieve this by multiplying the third row by -1 and adding it to the second row. Similarly, multiply the third row by 1 and add it to the first row. The augmented matrix after this operation is:
| 1 1 0 | 7 |
| 0 1 0 | 4 |
| 0 0 1 | 0 |
| 0 0 0 | 0 |
Step 7: Continue with the second row. Use row operations to make the entries above the leading 1 in the second column equal to 0. Multiply the second row by -1 and add it to the first row. The augmented matrix after this operation is:
| 1 0 0 | 3 |
| 0 1 0 | 4 |
| 0 0 1 | 0 |
| 0 0 0 | 0 |
The resulting row-echelon form of the augmented matrix represents the system of equations in a simplified form. From this, we can obtain the following system of equations:
x = 3
y = 4
z = 0
Therefore, the general solution to the given system of equations is:
x = 3
y = 4
z = 0