Testing for Linearity and Shift-Invariance I started school a week and a half ago and we have gotten already a big chunk of material and I wanted to check if what I am currently doing is correct. Here below are the problems to work out that help us for the exam in the future. (This class is signal processing)

for part (b)
Your demonstration of shift-invariance is not correct.
You time-shift the input signal by $n_0$, which is correct. But you also time-shift the sinusoidal, and this only happens if you were time-shifting the output.
The description of the system contains a signal that explicitly depends on time ($\sin(2\pi/7 n \cdots)$). In general this is a red-flag for non time-invariance.