Well you can do what you want but I don't think the path and stroke work that way.
What's probably happening is that you're selecting a 5 point stroke on one end and a 0 pt on the other, this just makes the whole path/line 0pt.
The path/line is made up of anchor points, which can expand into control lines for curves, these control lines have handles on them to control the line you've drawn and the next line that is going to be drawn.
The path/line usually has 3 basic handles/anchor points on it, at the beginning, middle and end for controlling the shape and angle of the path/line.
You add a weight/stroke to the path, but you can't have the path 10pt at one end and 0pt at the other.
You would need to have a closed path, one that is joined and completed.
To do what you want, draw a box of 10pt, or place guides 10pt apart.
Then with the pen tool, click 4 times, once at the top of the box (without the box selected, it's only a guide) then shift click at the bottom of the box/guide and then the third click should be where you want the line to be 0pt. Then click back where you started to close the path.
Now you move the widest end, where there are two anchor points and move them apart or closer together if needed. and you can move the other end, where it tapers out to 0 pt up or down, or left and right to control it.
Hope that helps.