Let any point on 3x-5y+ 1=0 be (t, (3t+ 1)/5), then the point symmetrical about y=x is ((3t+ 1)/5, t).

The graph ((3t+ 1)/5, t) determined by any point is a straight line 5x-3y- 1=0, which is the required proof.

Does the second question mean the minimum value of the sum of the distances from two points to the X axis?

That is to say, it is proved that the minimum distance AC+BC between two points A and B in the fourth quadrant and the moving point C on the X axis is that the position of point C is the intersection of the connecting line between point A' and point B on the X axis and the X axis.

First of all, it is clear that the shortest distance between two points is the direct connection between two points.

AC+BC=A'C+BC, connecting A'B, only when point c is on the line of A'B, A'C+BC=A'B, and the distance is the shortest.