What causes confusion in testing time invariance?

[Bullington]

I understand what time invariance means but there are a few catches that I'm completely confused about: Suppose we have $$y(t)=x(\alpha t-\beta)$$ to test time invariance we shift the input then "plug" it into the output:$$x_1(t_1)=x(t-t_o)$$ so this is when I become confused; when we plug $$x_1(t_1)$$ into $$y(t)$$ we get: $$x(\alpha t-t_o-\beta)$$ instead of $$y(t)=x_1(\alpha t_1 - \beta)=x(\alpha (t-t_o) -\beta)$$ I believe there is some underling math I am unaware of; but, what is it? Any sources?

 

[Simon Bridge]

... or the proposed mapping fails the test. ie. perhaps $y(t) = x(\alpha t - \beta)$ is not time invariant?
See examples: https://cseweb.ucsd.edu/classes/fa05/cse166/LSI.pdf