I'm not sure I understand the last bit "How many lines of slope 1 pass through at least one point of S?" .....

You can draw a line ofanydesired gradient (including m = 1) throughanygiven point (including points in S).

So I would've thought the answer would be equal to the number of points in S .....

Do you mean how many tangents to either xy = 2 or ....? Or, perhaps how many lines passing through two of the points in S ....?

Anyway, you certainly need to know how many points are in S ......

Solve .

Let , say:

.

Therefore and so there are two solutions for x. Therefore there are two points in S.