In this problem, $m$ and $n$ are integers.

If $m\equiv 4\pmod{14}$ and $m-n\equiv 10\pmod{14}$, then what is the remainder when $m+n$ is divided by $14$?
To solve for $n$, I can add $n$ to both sides and get $4\equiv n+10\pmod{14}$. Then I can subtract $10$ from both sides and get $-6\equiv n\pmod{14}$. But I prefer to have a positive remainder, so I can add $14$ to both sides and get $8\equiv n\pmod{14}$. This means that $n$ has the same remainder as $8$ when divided by $14$. Now I can use this to find $m+n$ modulo $14$. I can add the congruences for $m$ and $n$ and get $m+n\equiv 4+8\pmod{14}$. Simplifying, I get $m+n\equiv 12\pmod{14}$. This means that $m+n$ has the same remainder as $12$ when divided by $14$.

