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 26 miles. The second and third days she averaged 12 miles per day. The last two days she hiked a total of  miles. The total hike for the first and third days was 22 miles. How many miles long was the trail?

 Aug 27, 2023

Let d1​, d2​, d3​, and d4​ be the number of miles Phoenix hiked on days 1, 2, 3, and 4, respectively. We are given the following system of equations:

\begin{align*} d_1 + d_2 &= 26 \ d_2 + d_3 &= 24 \ d_3 + d_4 &= 22 \ d_1 + d_3 &= 22 \end{align*}Subtracting the second and third equations, we get d1​−d4​=2. Substituting this into the fourth equation, we get 2+d4​=22, so d4​=20. Substituting this into the third equation, we get d3​+20=22, so d3​=2. Substituting this into the second equation, we get d2​+2=24, so d2​=22. Substituting the values of d2​ and d3​ into the first equation, we get d1​+22=26, so d1​=4.

Therefore, Phoenix hiked a total of d1​+d2​+d3​+d4​=4+22+2+20=48​ miles on the Rocky Path Trail.

 Aug 27, 2023

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