I'm sure you could since the change in voltage for both circuits would still be the same so if you put both equations equal to each other, you could solve for i.
wait why would it be 1.565(r+5)? that's the current value for the open circuit. So shouldn't it be 1.636(r+5)? and when I solved that by setting both equations equal to each other, i got r=-41.8 which doesn't make any sense. And when I tried solving it with your equations, I got r=33.45.
I know the answers are 9 V and .5 ohms but I have no idea how to get there. Originally I solved for Delta V in the open circuit through (1.636)(5) which gave me 8.18V. Then, I solved for Delta V in the closed circuit through (1.565)(1/((1/5)+(1/10)) which gave me 5.51 V. I used these voltages...