Question 113520
Your analysis is correct and you're on the right track.
You made a mistake at this point.

{{{   10x + y = 10y + x + 3}}}
The new number (digits reversed) increased by 3 equals 4 times the original number.
You forgot the 4.
{{{   highlight(4)(10x + y) = 10y + x + 3}}}
Now together with 
{{{x+y=7}}}
should get you a quick answer. 
Good luck.
Post another question if you get stuck.