Question

Miguel bikes 23 km per hour and starts at mile 10. Gabby bikes 28 km per hour and starts at mile 0. Which system of linear equations represents this situation?

Answer 1

To represent the situation of Miguel and Gabby biking using a system of linear equations, we can start by defining variables for time and distance.

Let:
- \( t \) = time in hours
- \( d_M \) = distance traveled by Miguel in miles
- \( d_G \) = distance traveled by Gabby in miles

### Miguel's Situation:
- Miguel starts at mile 10. His speed is 23 km/h.
- To convert km/h to miles per hour (mp/h), we use the conversion factor:
\[
1 \text{ km/h} \approx 0.621371 \text{ miles/hour}
\]
Therefore, Miguel's speed in miles per hour:
\[
23 \text{ km/h} \times 0.621371 \text{ miles/km} \approx 14.291 \text{ miles/hour}
\]

- The equation for Miguel can be expressed as:
\[
d_M = 10 + 14.291t
\]

### Gabby's Situation:
- Gabby starts at mile 0. Her speed is 28 km/h.
- Converting her speed to miles per hour:
\[
28 \text{ km/h} \times 0.621371 \text{ miles/km} \approx 17.398 \text{ miles/hour}
\]

- The equation for Gabby can be expressed as:
\[
d_G = 0 + 17.398t \quad \text{or simply} \quad d_G = 17.398t
\]

### System of Equations:
Now we can write the system of equations representing the distances traveled by Miguel and Gabby over time \( t \):
\[
\begin{cases}
d_M = 10 + 14.291t \\
d_G = 17.398t
\end{cases}
\]

This system can be used to analyze their distances over time as they bike towards the same goal.