**Theorem: Echelon Form and Consistency**

A linear system is consistent if and only if the rightmost column of its augmented matrix is NOT a pivot column in the echelon form.

**Statement:** Given an augmented matrix `[A | b]`, the system `Ax = b` is consistent if and only if the echelon form of `[A | b]` has no row of the form:
```
[ 0  0  ...  0  |  c ]
```
where `c ≠ 0`.

**Explanation:** Such a row corresponds to the equation `0 = c` (with `c ≠ 0`), which is a contradiction. This happens precisely when the last column (the constants) becomes a pivot column during row reduction.

**Procedure to check consistency:**
1. Row reduce the augmented matrix to echelon form
2. Examine each row: if any row has all zeros in the coefficient part but a nonzero entry in the constant column, the system is inconsistent
3. Otherwise, the system is consistent

**Corollary:** A homogeneous system `Ax = 0` is always consistent (the trivial solution `x = 0` always exists), since the last column is always zero and can never become a pivot.