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I want to know how to solve those types of problems.. is it by inspection ?

Consider the linear system below. When the inputs to the system $x_1[n]$, $x_2[n]$ and $x_3[n]$, the responses of the systems are $y_1[n]$, $y_2[n]$ and $y_3[n]$ as shown.

enter image description here

a. Determine whether the system is time invariant or not. Just your answer.

b. What is the impulse response?

Edit: Assuming a general case where the given inputs don't contain a scaled impulse like $x_2[n]$

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Just to make sure: $n$ represents steps in time? – Ron Gordon Jan 3 '13 at 14:45
yes, $n$ represents steps in time – Belbesy Jan 3 '13 at 14:47

You should use linearity to calculate the response to unit impulses at $-2$, $0$, and $2$. Let $\delta(n)$ be the unit impulse at time $n$. Since $\delta(2)$ can be written as $(x_2-x_1)/2$, we have $$ T(\delta(2))=T((x_2-x_1)/2)=(y_2-y_1)/2=[0,0,1/2,-1,-1/2] $$ Similarly, $$ T(\delta(0))=T(x_2/2)=y_2/2=[0,0,1/2,-1/2,0] $$ ... and $T(\delta(-1))$ can be found in much the same way. If the system were time-invariant, then $T(\delta(2))$ would be $T(\delta(0))$ delayed by two units of time. This is clearly not the case.

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