A system is time invariant if for any time shifted input signal the system produces a shifted output such that if an input $ x(t) $ produced an output $ y(t) $ then the input $ x(t + t_0) $ would produced the output $ y(t + t_0) $
Time Invariant Signal
$ S_1(t) = 5x(t) $
$ S_2(t) = t - t_0 $
$ x(t) -> S_1(t) -> S_2(t) -> 5x(t - t_0) $
$ x(t) -> S_2(t) -> S_1(t) -> 5x(t - t_0) $
This means that $ S_1(t) = 5x(t) $ is a time invariant signal.
Time Variant Signal
$ S_1(t) = x(1-t) $
$ S_2(t) = t - t_0 $
$ x(t) -> S_1(t) -> S_2(t) -> 5x(1 - t + t_0) $
$ x(t) -> S_2(t) -> S_1(t) -> 5x(1 - t - t_0) $
This means that $ S_1(t) = 5t $ is a time invariant signal.
