Lets label the planes:
A : x - z = 1
B : y + 2z = 3
C : x + y - 2z = 1
You found one point in the intersection of A and B. Call this point P.
Now how many planes go through point P and is perpendicular to C? Infinitely many.
You used the vector <1, 0, 1/2> and that gives one solution, but think about rotating <1,0,1/2> in the plane of C. Any of these rotated vectors gives you another possible solution.
The solution is incorrect because you need to use the fact that the intersection of A and B is a line. So the solution should pass through at least 2 points in the intersection of A and B. This will reduce the number of possible planes perpendicular to C to just one.