Answer
Inconsistent.
Work Step by Step
Express x from the first equation.
$x+2y-z=1 \rightarrow x=-2y+z+1$
$2x+3y-4z=-3$
$3x+6y-3z=4$
Substitute $ x=-2y+z+1$ into the second and the third equation.
$2(-2y+z+1)+3y-4z=-3$
$3(-2y+z+1)+6y-3z=4$
Simplify.
$-y-2z=-5$
$3=4$
After simplification of the equations we got that $3=4$ which is false. No values of x, y and z satisfy this equation and therefore the system is inconsistent.