Answer
Inconsistent.
Work Step by Step
Express x from the second equation.
$2x+3y-z=1$
$x+2y=3 \rightarrow x=-2y+3$
$x+3y+z=4$
Substitute $ x=-2y+3$ into the first and the third equation.
$2(-2y+3)+3y-z=1$
$-2y+3+3y+z=4$
Simplify.
$-y-z=-5$
$y+z=1$
Add the equations.
$0=-4$
After addition of the equations we got that $0=-4$ which is false. No values of x, y and z satisfy this equation and therefore the system is inconsistent.