Converting vector form of line in R3 to general form
In $$\Rightarrow z+y-x = -3$$
You have not solved anything. What you have done is to eliminate $t$ through addition. This plane is one of infinitely planes on which the stated line lies.
Doing things differently, for example, I can get:
$z-2y-3x = -5$
by multiplying the second equation by $-2$ and the first one by $-3$, then summing everything.