Time invariace of a linear system dependent on a particular time instant

Question

$$y[n]=x[n]+35*x[n-1]+x[0]$$

Is this system time invariant? I am under the impression that $x[0]$ can be considered a constant. Am I right?

Answer 1

A time-invariant system reacts to a shifted input by an equal shift in its response, i.e. if $y[n]$ is the system's response to $x[n]$, then for the system to be time-invariant, its response to $x[n-k]$ must be $y[n-k]$ for some integer $k$.

Now let's assume that $y[n]$ is the response to $x[n]$ and let's see what happens if we use $x_1[n]=x[n-k]$ as an input signal:

$$y_1[n]=x_1[n]+35x_1[n-1]+x_1[0]=x[n-k]+35x[n-k-1]+x[-k]$$

which is not equal to $y[n-k]$, so the system is not time-invariant.

Answer 2

I'm not sure what you mean by "$x[0]$ can be considered a constant".

At any rate, this system is definitely not time invariant. Try feeding in $x[n] = \delta[n]$, shifting it over time, and seeing what happens.

In particular, putting in $x[n-1]$ does not give you $y[n-1]$, which is the property that defines time-invariance.