System of Equations

Question

Suppose you have entered a 82 mile biathlon that consists of a run and bicycle race. During your run, your average velocity is 8 miles per hour, and during your bicycle race, your average velocity is 21 miles per hour. You finish the race in 7 hours. What is the distance of the run? What is the distance of the bicycle race?

Answer 1

Let t_r = time of the run and t_b = time of the bicycle race

We are given that the total time of the race is 7 hr, so

(1) t_r + t_b = 7

We can write the equation for the total distance of the race as

(2) 8t_r + 21t_b = 82

If we multiply the first equation by –8 to get

–8t_r – 8t_b = –56

and add this to the second equation

8t_r – 8t_r + 21t_b – 8t_b = 82 – 56

we get

13t_b = 26 which means that t_b = 2 hr

and t_r = 7 – t_b = 7 – 2 = 5 hr

Answers

The distance of the run is 8 mi/hr × 5 hr = 40 mi

The distance of the bicycle race is 21 mi/hr × 2 hr = 42 mi

Answer 2

In order to answer this question, you first need to define some terms:

r = # of miles run

b = # of miles biked

EQUATION SETUP:

I know that the number of miles run added to the number of miles biked must add up to the total race miles:

r + b = 82

I also know that to find the amount of hours spent running I need to divide the number of miles run by 8 since I run 8mph (for example: 16mi / 8mph = 2hr), and do the same for number of miles biked and bike speed. So:

7 = r/8 + b/21

TO SOLVE: (SUBSTITUTION STRATEGY)

From there, the simplest way to isolate r or b is to solve r + b = 82 for b (or r but I will solve for b):

b = 82 - r

And then substitute the value of b into our other equation:

7 = r/8 + (82 - r)/21

SOLUTION:

From there, you solve by multiplying all terms in the equation by 8 and 21 to eliminate the denominators...

7*8*21 = r*21 + (82 - r)*8

Multiply and distribute...

1176 = 21r + 82*8 - 8r

Combine like terms and multiply...

1176 = 13r + 656

Combine like terms again...

520 = 13r

Divide by 13 to solve for r...

40 = r

Plug 40 in for r in our equation for b (b = 82 - r)...

b = 82 - 40 = 42

ANSWER:

The distance of the run is 40 miles, and the distance of the bicycle race is 42 miles.