Try

    Solve[Det[({{1, x, y}, {1, a, b}, {1, c, d}})] == 0, y][[1]] //Collect[#, x, Simplify] &
    (*{y -> (-b c + a d)/(a - c) + ((b - d) x)/(a - c)}*)

create equation

    (% /. Rule -> Equal) [[1]]
    (*y == (-b c + a d)/(a - c) + ((b - d) x)/(a - c)*)