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To show that if a system of linear equations is consistent, it can be reduced to echelon form, we will follow a proof by contradiction: 1. Assume that a linear system is consistent but cannot be reduced to echelon form. 2. Since the given system is consistent, there is at least one solution to the system of equations. 3. If there is at least one solution, then using Gaussian Elimination or another row reduction technique, we can attempt to reduce the matrix to echelon form. 4. If we fail to reduce the matrix to echelon form, that means at some point, we encountered a row where all elements are zero except for the constant term, i.e., a degenerate equation with a nonzero constant. 5. The existence of a degenerate equation with a nonzero constant contradicts our assumption that the system is consistent, as no such equation can have a solution. 6. Therefore, our assumption was incorrect, and a consistent system of linear equations must be reducible to echelon form.

To show that if a system of linear equations can be reduced to echelon form, it is consistent, we will follow these steps: 1. Assume that a linear system can be reduced to echelon form. 2. If a system can be reduced to echelon form, that means no degenerate equation with a nonzero constant exists in the system (otherwise, it would not be possible to achieve echelon form). 3. In echelon form, we can use back-substitution to find the solutions to the system of equations. 4. Since we can obtain solutions using back-substitution, we can conclude that the system of linear equations is consistent. In conclusion, the system of linear equations is consistent if and only if it can be reduced to echelon form.