If I get the slope of a line, and one point that is on it, then, are they define exactly ONE line?
The point–slope form of linear equations ($y - y_1 = m( x - x_1 )$) need only the slope and the coordinates of that given point. And this equation describes exactly one line.
But I'm not sure that is it enough?
Yes, that is enough. You can determine unambiguously whether a point is on the line from the equation. There is a problem with vertical lines in this form, however. You just need to use $x=x_1$ for that.