A linear relation represents a "constraint" between two variables x and y (two degrees of freedom) is ax + by + c = 0 (a,b,c are constants). There is no need to keep three parameters, two are enough, for instance, dividing by b (b0), ux + y + v = 0 (u = a/b, v = c/b)
x,y,z were reserved for "point coordinates" while u,v,w were used for "line-coordinates". For instance, a straight line in the x,y plane is given by the equation y + ux + v = 0. This equation gives the united position of the line (u,v) and the point (x,y), i.e., the point lies on the line and the line goes through the point. So, (-u) is the trigonometric tangent of the angle that the line makes with the x-axis and (-v) is the y-intercept. In the dual representation (interchanging line and point coordinates), the family of straight lines through the point (x,y) is given by v + xu + y = 0. So, given x and y, the equation represents the linear relation between "slope" and "y-intercept".