Let the number of miles hiked on the first day be $a$, the number of miles hiked on the second day be $b$, and the number of miles hiked on the third day be $c$. We are given three equations based on the given information: $$a+b=22,$$$$b+c=20,$$$$a+c=26.$$
We are trying to find the total number of miles hiked, which is $a+b+c$. Notice that if we add the three equations above, we get $2a+2b+2c=68 \implies a+b+c=\boxed{34}$.
Phoenix hiked the Rocky Path Trail last week. It took four days to complete the trip. The first two days she hiked a total of $22$ miles. The second and third days she averaged $28$ miles per day. The last two days she hiked a total of $20$ miles. The total hike for the first and third days was $26$ miles. How many miles long was the trail?
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