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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)

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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.

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