Question

Tom bikes 20km per hour and starts at mile 5. Alex bikes 25 km per hour and starts at mile 0. Can you find the system of linear equations that represents this situation?

Answer 1

To represent the situation with a system of linear equations, we can set up equations for the positions of Tom and Alex as functions of time.

Let \( t \) be the time in hours.

**For Tom:**
- Tom starts at mile 5 and bikes at a speed of 20 km/h.
- His position \( P_T \) as a function of time can be represented as:
\[
P_T(t) = 5 + 20t
\]

**For Alex:**
- Alex starts at mile 0 and bikes at a speed of 25 km/h.
- His position \( P_A \) as a function of time can be represented as:
\[
P_A(t) = 0 + 25t = 25t
\]

We can now write the system of linear equations representing the positions of Tom and Alex over time:

1. \( P_T(t) = 5 + 20t \) (Tom's position)
2. \( P_A(t) = 25t \) (Alex's position)

If we are interested in when they meet, we can set \( P_T(t) \) equal to \( P_A(t) \):
\[
5 + 20t = 25t
\]

This simplifies to:
\[
5 = 25t - 20t
\]
\[
5 = 5t
\]
\[
t = 1
\]

At \( t = 1 \) hour, they will meet. The system of equations remains as given above to represent their positions:

1. \( P_T(t) = 5 + 20t \)
2. \( P_A(t) = 25t \)

This completes the formulation of the linear equations based on the biking situation of Tom and Alex.