The system is consistent and has infinitely many solutions.

Parameter: \(\displaystyle{x}{3}∈{R}\)

Row 2: \(\displaystyle{x}{2}+{3}{x}{3}={1}\to{x}{2}={1}-{3}{x}{3}\)

Row 1: \(\displaystyle{x}{1}-{2}{x}{3}={6}\to{x}{1}={6}+{2}{x}{3}\)

Infinitely many solutions,

\(\begin{cases}x_1=6+2x_3\\x_2=1-3x_3\end{cases}, x_3 ∈ R\)