(The book is :Paul Bamberg, ‎Shlomo Sternberg - A Course in Mathematics for Students of Physics Vol.1 . Pag 47 es 1.2)

An affine function of a line is
and represent the real number that, multiplied for a basis and starting from an origin of the line gives a certain point of the line, so a origin of the line and a basis is implicitly taken when defining the affine coordinate function.

Suppose we have 2 affine lines (NOT paralells), for each line we set an arbitrary origin and an arbitrary basis. I choose a point P in the first line and i get x(P) = 3. Then, I choose an arbitrary point P' in the second line , and i get y(P') = 1 . So, the mapping beetween the first line and the second line is F(α)=α/3.

Why it's wrong and F(α) should be F(α)=rα+s, and not just F(α)=rα as my result ?