# is y[n] = x[n] + n time invariant?

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?

$$y[n] = x[n] + n$$ 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
$$y_2[n] \neq y[n-k]$$.