How to set up and solve a biathlon time and distance system

Key concept: This kind of word problem is really just a translation exercise. Once the two variables are named, the equations almost build themselves.

Step by step

Let:

• $r$ = number of hours Becky runs
• $b$ = number of hours Becky bikes

Step 1: Write the time equation

The total time is 12 hours: $r+b=12$

Step 2: Write the distance equation

She runs at 6 mph and bikes at 20 mph, covering 198 miles total: $6r+20b=198$

Step 3: Solve the system

From the first equation: $r=12-b$ Substitute into the distance equation: $6(12-b)+20b=198$ $72-6b+20b=198$ $72+14b=198$ $14b=126$ $b=9$ Then $r=12-9=3$

Answer

• Becky runs for 3 hours
• Becky bikes for 9 hours

Pitfall alert

Don’t swap the speeds. A small label mistake like putting 20 on running and 6 on biking changes the entire system. Also, make sure the hours add to 12 before checking the distance equation; that’s a fast way to catch sign or setup errors.

Try different conditions

If the total distance had been different, the system would still use the same structure: one equation for time and one for distance. If the speeds were changed, the coefficients in the distance equation would change, but the solving process would be identical.

Further reading

linear system, distance-rate-time, substitution