Let R2 denote the usual (x, y)-plane. We can define a partition of R2 by using the
lines with slope 2. Describe the corresponding equivalence relation ∼ by giving the
conditions on the coordinates so that (x1, y1) ∼ (x2, y2).
I have we must have (y2-y1)/(x2-x1)=2
Okay, your equivalence classes are the individual lines. Two points are equivalent, if and only if they lie on the same line. That is, as you say, which is the same as saying that and the same as . But I cannot see how you got " " from that! Any one of those first three equations will do. The simplest, in my opinion, is the first: if and only if .