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My steps were as follows:

$\ x_2[n] = x[n-k] $

$\ y[n-k] = x[n-k] + (n-k) $

and

$\ y_2[n] = x_2[n] + n = x[n-k]+(n-k)$

Does this mean that it is indeed time invariant?

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No; the system given by

$$ y[n] = x[n] + n $$ is time-varying, due to the added term $n$.

Your mistake is in the line : $$\ y_2[n] = x_2[n] + n = x[n-k]+(n-k) $$ which should be instead

$$\ y_2[n] = x_2[n] + n = x[n-k]+n$$ and therefore implies that

$$y_2[n] \neq y[n-k] $$.

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