If the cascade X(t) -> time delay by t0 -> system -> y(t) yields the same output as the cascade X(t) -> system -> time delay by t0 -> z(t) for any t0, then the system is called time invarient.
E.g.
y(t) = 2X(t) is time invariant
Since for X(t) -> time delay by t0 -> system -> y(t)
z(t) = 2y(t)
and y(t) = X1(t-t0)
=> z(t)=2X1(t-t0)
And for X(t) -> system -> time delay by t0 -> z(t)
z(t) = y(t-t0)
and y(t) = 2X2(t)
=> z(t)=2X2(t-t0)
yields the same output of z(t)
y(t)=x(1-t) is not time invariant
Since for X(t) -> time delay by t0 -> system -> y(t)
z(t) = y(1-t)
and y(t) = X1(t-t0)
=> z(t)=X1(1-t-t0)
And for X(t) -> system -> time delay by t0 -> z(t)
z(t) = y(t-t0)
and y(t) = X1(1-t)
=> z(t)=X1(1-(t-t0))=X1(1-t+t0)
yields different output of z(t)
