### Instruction: 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 28 miles. The total hike for the first and third days was 22 miles. How many miles long was the trail?### Response: This problem involves finding the sum of four unknown distances: the first day's hike, the second day's hike, the third day's hike, and the fourth day's hike.
I can use the given information to set up a system of four equations with four variables.
Let x be the first day's hike, y be the second day's hike, z be the third day's hike, and w be the fourth day's hike. Then I have the following equations: x + y = 26, y + z = 24, z + w = 28, and x + z = 22.
I can solve this system by using substitution or elimination.
I notice that the first equation gives me y in terms of x, so I can substitute that into the second equation to get z in terms of x. Then I can substitute that into the third equation to get w in terms of x. Then I can substitute that into the fourth equation to get x.
So, starting from the first equation, I get y = 26 - x.
Substituting that into the second equation, I get 26 - x + z = 24, or z = x - 2.
Substituting that into the third equation, I get x - 2 + w = 28, or w = 30 - x.
Substituting that into the fourth equation, I get x + x - 2 = 22, or 2x = 24, or x = 12.
Now that I have x, I can use the other equations to find y, z, and w.
From the first equation, I get y = 26 - x = 26 - 12 = 14.
From the second equation, I get z = x - 2 = 12 - 2 = 10.
From the third equation, I get w = 30 - x = 30 - 12 = 18.
Now I have the values of x, y, z, and w, which are the distances for each day's hike.
To find the total distance of the trail, I just need to add them up.