Irene, in my partial solution you are not solving a single equation. You need to solve the system of equations simultaneously. That is, you are looking for digits A, B and C that satisfy all 3 equations.
There are many ways to solve a system of equations: elimination, substitution and Gauss Jordon Reduction would be the best in this example.
I think substitution is actually quickest here. Let me do it that way as an example for you:
C+B=A+10
B+C+1=C+10
A+A+1=B
From the second equation, we immediately get B=9.
Substituting this into the third equation we get
2A+1=9
2A=8
A=4
Now substituting A=4 and B=9 into the first equation gives the following:
C+9=4+10
C+9=14
C=5