Question 300399
The original number is AB where the actual value is 10*A+B.
The reversed digits number is BA and its actual value is 10*B+A.
Nw turn the words into equations. 
"sum of digits is 11"
1.{{{A+B=11}}}
"the digits of a two-digit number are reversed,the new number is 9 more than the original number"
{{{10*B+A=9+10*A+B}}}
{{{-9A+9B=9}}}
2.{{{-A+B=1}}}
Now you have two equations in A and B.
Add them together to eliminate A and solve for B. THen go back and solve for A. 
To check your answer, run through the two requirements and make sure the digits meet them.