Question

Write a system of equations to model each situation. Then, identify a solution.

On Sunday, Bonnie ran a 20-mile race in 2.3 hours. The route for the race goes over hills, and she knows she can run 8 minutes per mile uphill and 6.5 minutes per mile downhill. How long did she spend running uphill and downhill?

Answer 1

To model Bonnie's race, we can set up a system of equations using x as the number of miles she ran uphill and y as the number of miles she ran downhill. Using the given information, we can write the equations x + y = 20 and 8x + 6.5y = 2.3. By solving this system, we find that there is no valid solution.

Step-by-step explanation:

To solve this problem, we can set up a system of equations to represent Bonnie's race. Let x represent the number of miles she ran uphill, and y represent the number of miles she ran downhill. Since she ran a total of 20 miles, we can write the equation x + y = 20. Additionally, since she ran 8 minutes per mile uphill and 6.5 minutes per mile downhill, we can write the equation 8x + 6.5y = 2.3, representing the time it took her to complete the race. To solve this system, we can use substitution or elimination.

Substitution Method:

1. Rewrite the first equation as x = 20 - y.
2. Substitute 20 - y for x in the second equation: 8(20 - y) + 6.5y = 2.3.
3. Simplify and solve for y: 160 - 8y + 6.5y = 2.3 => -1.5y = -157.7 => y ≈ 105.13.
4. Substitute the value of y back into the first equation to solve for x: x = 20 - 105.13 ≈ -85.13.

Since we can't have a negative distance, the solution is not valid in this case. Therefore, there is no solution to this system of equations.