Answer
Yes.
Ordered triple $(-1,3,1)$ satisfies all three equations,
$x+y+z = 3$
$-x+y+z = 5$
$x+2y-3z =2$
So, $(-1,3,1)$ is a solution.
Work Step by Step
Substituting $x=-1$, $y=3$, $z= 1$ in
$x+y+z = 3$
$-x+y+z = 5$
$x+2y-3z =2$,
$x+y+z = 3$
$-1+3+1=3$
$3 = 3$
$-x+y+z = 5$
$-(-1)+3+1= 5$
$1+3+1= 5$
$5=5$
$x+2y-3z =2$
$-1+2(3)-3(1)= 2$
$-1+6-3 = 2$
$2 = 2 $
Ordered triple $(-1,3,1)$ satisfies all three equations. So, $(-1,3,1)$ is a solution.