I'm stuck on a time-variance question in my homework and I think it has something to do with my understanding of how time shifting functions works. The problem asks to determine whether the system defined below is time-variant or time-invariant. $$ S: x(t) \to y(t) = x(t-2) + x(2-t) \tag{1}$$ I solved it such that $$ x_{delayed}(t) = x(t - \tau)\tag{2}$$ and $$ y_{delayed}(t) = x(t-\tau-2) +x(2-t +\tau) \tag{3}$$
Then I delayed the original output $ y(t) $ from $(1)$ to $$y(t-\tau) = x(t-\tau-2) + x(2-t+\tau) \tag{4}$$ which I believe is incorrect. I think that instead, it should be $$y(t-\tau) = x(t-\tau-2) + x(2-t-\tau) \tag{5}$$
I believe my problem comes from my understanding of how to delay a function related to another function. Searching around didn't help much. Ultimately my question is, should $$ y(t) = x(-t) \tag{6}$$ $$ y(t - \tau) = x(-t + \tau\tag{7})$$ or $$ y(t-\tau) = x(-t-\tau)\tag{8}$$ Using $(7)$ gives my my current answer, $(4)$, while using $(8)$ gives me what I believe is the correct answer, $(5)$. I have found proofs for both answers which only confuses me more, so which is correct and how should I time shift the function? Any help is welcome and appreciated, thanks.