To find the augmented matrix for each system of linear equations, we can write the coefficients of the variables and the constants in a matrix form.
(a) For the system of linear equations:
3x1 − 2x2 = −1
4x1 + 5x2 = 3
7x1 + 3x2 = 2
We can write it in matrix form as:
| 3 -2 | -1 |
| 4 5 | 3 |
| 7 3 | 2 |
The augmented matrix for this system is the matrix on the left-hand side of the equation, augmented by the matrix on the right-hand side.
(b) For the system of linear equations:
2x1 + 2x3 = 1
3x1 − x2 + 4x3 = 7
6x1 + x2 − x3 = 0
We can write it in matrix form as:
| 2 0 2 | 1 |
| 3 -1 4 | 7 |
| 6 1 -1 | 0 |
The augmented matrix for this system is the matrix on the left-hand side of the equation, augmented by the matrix on the right-hand side.
(c) For the system of linear equations:
x1 + 2x2 - x4 + x5 = 1
3x2 + x3 - x5 = 2
x3 + 7x4 = 1
We can write it in matrix form as:
| 1 2 0 -1 1 | 1 |
| 0 3 1 0 -1 | 2 |
| 0 0 1 7 0 | 1 |
The augmented matrix for this system is the matrix on the left-hand side of the equation, augmented by the matrix on the right-hand side.
(d) For the system of linear equations:
x1 = 1
x2 = 2
x3 = 3
We can write it in matrix form as:
| 1 0 0 | 1 |
| 0 1 0 | 2 |
| 0 0 1 | 3 |
The augmented matrix for this system is the matrix on the left-hand side of the equation, augmented by the matrix on the right-hand side.
I hope this explanation helps you understand how to find the augmented matrix for a system of linear equations. Let me know if you have any further questions!