Final answer:
An augmented matrix indicates the number of solutions to the linear system of equations. By performing row operations to reduce the augmented matrix, we can determine whether the system has a unique solution, infinitely many solutions, or no solution.
Step-by-step explanation:
When given an augmented matrix, it indicates the number of solutions to the linear system of equations. The number of solutions can be determined by performing row operations to reduce the augmented matrix to its row echelon form or reduced row echelon form.
If all the rows have a leading 1 (pivot) and there are no rows of zeros except for the last row, then the system has a unique solution. If there is a row of zeros, then the system has infinitely many solutions. If there is a row of zeros with a nonzero entry in the last column, then the system has no solution.
For example, if we have the augmented matrix:
[1 0 2 | 5]
[0 1 -3 | 2]
This can be simplified to:
[1 0 2 | 5]
[0 1 -3 | 2]
Since all the rows have a leading 1 and there are no rows of zeros except for the last row, the system has a unique solution, x = 5 and y = 2.