Hello, I'm trying to figure out the answer to the following question regarding finite affine planes:

My intuition is that there are far too many lines for this to be a finite affine plane. We'd expect there to be n^2+n = 6^2+6 = 42 lines, but when I started computing them there seemed to be more than this.Let a set P of points consist of all ordered pairs of elements from Z6.

Define a ‘line’ to consist of all solutions (x, y) to an equation of the form ax + by = c, where a, b, c 2 Z6, at least one of a and b is nonzero, and addition and multiplication are carried out modulo 6. For example, x + y = 1 defines the ‘line’ {(0, 1), (1, 0), (2, 5), (3, 4), (4, 3), (5, 2)}. Let L be the set of all ‘lines’.

Determine whether (P,L) is a finite affine plane, and prove your answer.

Is there a more efficient method for determining this than working out all the lines?

Thank you.