Given 3 points: $A (1,2,3) B (1,1,2) C (2,1,2)$
I have calculated the equation of the vector plane to be $z-y=1$.
However, I do not understand how the equation of the plane can be independent of $x$ when $x$ is clearly changing in the coordinates above. I feel I am missing something crucial. Thanks in advance.
You determined (correctly) that the plane that contains $A$, $B$, and $C$ is the plane $z-y=1$. What this means is that the points of the plane are all the points of the form $(x,y,y+1)$. As you can see, $x$ can, in fact, take any value you want. So, the fact that $A$, $B$, and $C$ don't have all the same first coordinate is not a problem.