Detect an inconsistent overdetermined system

Use columns $x,y,z$ and the right-hand constant. In the second row the missing $y$ term has coefficient zero. The source’s displayed augmented matrix is

$\left[\begin{array}{ccc|c}1&2&-2&3\\1&0&3&-6\\5&3&-1&2\\3&-2&1&0\end{array}\right].$

The printed calculator screen reads NO SOLUTION FOUND. The following elimination explains why this is an inconsistency rather than an entry error.

Label the equations (1)–(4). From (1)−(2), $2y-5z=9$ (5). From $5\times$(2), $5x+15z=-30$ (6). Subtract (6) from (3): $3y-16z=32$ (7). Then (7)−$\tfrac32$(5) gives $-\tfrac{17}{2}z=\tfrac{37}{2}$, so $z=-37/17$. Equations (5) and (6) give $y=-16/17$ and $x=9/17$.

None of the preceding steps used (4). At the only candidate, its left side is $3(9/17)-2(-16/17)+(-37/17)=22/17$, whereas its right side is $0$. Since $22/17\ne0$, no triple satisfies all four equations. A system can have no solution, one unique solution or infinitely many solutions. The latter two are consistent; no solution means inconsistent.