Just wondering how one derives the equation of a line y=mx+b from this general expression f(x,y)=a+bx+cy. I believe this expression represents plane geometry.
Pick one point on the plane, say and other one . We will try to see the relation between and Now impose that they are in a line:
Let be the angle between the black and the red segments. We have where and . Then
f(x,y)= z= ax+ by+ c is the equation of a plane in three dimensions. What do you mean by "derive the equation of a line"? Which line? There exist an infinite number of lines in that plane. We could, for example, get the equation of the line where it intersects the xy plane by setting z= 0: 0= ax+ by+ c which is the same as -by= ax+ c and then divide both sides by -b: y= (-a/b)x- (c/b) which is of the form "y= mx+b" with m= -a/b and "b"= -c/b.