Question

There are two trails near Lexi's house that she runs regularly, a short loop and a long loop. Last week, she ran 2 short loops and 6 long loops, for a total of 48 kilometers. This week, she ran 2 short loops and 2 long loops, covering a total of 20 kilometers. What is the length of each loop? The short loop has a length of kilometers, and the long loop has a length of kilometers.

A) Short loop: 4 kilometers, Long loop: 6 kilometers
B) Short loop: 6 kilometers, Long loop: 4 kilometers
C) Short loop: 8 kilometers, Long loop: 5 kilometers
D) Short loop: 5 kilometers, Long loop: 8 kilometers

Answer 1

Using the system of linear equations derived from Lexi's running patterns, it turns out the short loop is 3 kilometers and the long loop is 7 kilometers. This does not match any of the provided multiple-choice options, indicating there may be an error in the question itself.

Step-by-step explanation:

To determine the length of each loop that Lexi runs, we can set up two equations based on the given information and solve the system of linear equations.

Let's denote:
S = length of the short loop
L = length of the long loop

From the first week:
2S + 6L = 48 (1)

From the second week:
2S + 2L = 20 (2)

Now, we can solve these equations step-by-step:

1. Divide equation (2) by 2:
   S + L = 10
2. From this, we can express S in terms of L:
   S = 10 - L (3)
3. Substitute S from equation (3) into equation (1):
   2(10 - L) + 6L = 48
4. Solve for L:
   20 - 2L + 6L = 48
   4L = 28
   L = 7
5. Substitute L back into equation (3) to find S:
   S = 10 - 7
   S = 3

Based on these calculations, the short loop is 3 kilometers and the long loop is 7 kilometers, which is not one of the provided options. It appears there might be an error in the question since none of the given options match the solution derived from the provided information.