Hi, I have two solutions to a problem one devised by me and the other by the instructor. They have the same result, but which is actually done correctly?

Problem:
We have an initial location x_{init}=1000 with an uncertainty modelled with a Gaussian with \sigma_{x}^2=900. We take a GPS measurement z=1100 which has an error variance \sigma_{z}^2=100.

Write the probabibility density functions of the prior p(x) and the measurement p(z|x). And using Bayes rule, what is the posterior p(x|z)?
Solution 1:
p(x) = N(x; 1000, 900)
p(z|x) = N(z-x; 0, 100)
p(x|z) = \frac{p(z|x)p(x)}{p(z)}
p(z) = \int p(z|x)p(x) = N(z; 1000, 1000)
\vdots
p(x|z) = N(x; 1090, 90)
Solution 2:
p(x) = N(x; 1000, 900)
p(z|x) = N(z; 1100, 100)
p(x|z) = \frac{p(z|x)p(x)}{p(z)}
p(z) = \int p(z|x)p(x) = \eta (a normalizing constant)
\vdots
p(x|z) = N(x; 1090, 90)
The specific calculations are not important, but I hope I've given enough information to say which p(z|x) is actually correct. Or maybe they both are, or neither?