Time Invariant Systems are systems that produce the same output for a given input signal that has a time shift as it would without the time shift except for that the output will be based on the time corresponding to the input function's time shift.
An example of a time-invariant system would be one represented by the equation y(t) = x(t + 1) where if you change y(t) to be y(t+1) then x(t + 1) will become x((t+1) + 1). You can prove that this system is time invariant by inputing the signal x(t) into a time delay system and then inputing it into the system and comparing the results to imputing the signal x(t) into the system and then inputing that results into the time-delay circuit and seeing if the results are the same. If the two final outputs of either cascade are the same then the system is time-invariant.
