Depressing the cubic, substitution

Question

I have been reading about Cordano's solution, tschirnhaus transformation, etc, for the past few days and one thing that I cannot understand, if anything, is the substitution of y=x-3/a or x=y+3/a (if the coefficient in front of cubed x is 1) . I have found pretty good lectures with Professor William Dunham and other links here and in other places, but none go into any details regarding that substitution. I want to know if there is any general form for it, for it feels very specific and since I don't want to just memorize it I would love to have some material (or any explanation here) to read but my searching lead me nowhere, either it has another popular name I am unaware of or there is nothing, which is unlikely. What little I know is that it is used to remove n-1 degree term to ease calculation but I want to read how it was derived or something similar, a read that isn't too advanced. I want to depress a cubic that has a square term in it, but actually know what or how the substitution that I used came about. Thank you!