Three-Legged Race: System of Equations Problem

What This Problem Teaches

• Setting up a system of equations from a real-world scenario with multiple moving objects
• Understanding how different starting positions and rates create linear relationships
• Solving systems by substitution and interpreting the intersection point graphically
• Connecting algebraic solutions to coordinate geometry and visual representation
• Distinguishing between distance traveled and position from a reference point

Let's Draw It

This coordinate graph shows both runners' positions over time. The adults start at the origin (0,0) with a steeper slope representing their faster speed. The kids start 4 meters ahead with a gentler slope. The intersection at (2, 8) tells us they meet after 2 seconds, when both are 8 meters from the adult starting line.

Solution: Method 1 — Position Equations Approach

Step 1 — Define variables and establish reference point

Let t = time in seconds from when both teams start moving. Let's measure all distances from the adults' starting line as our reference point (position = 0 for adults at t = 0).

Step 2 — Write position equations for each team

Since position = starting position + (rate × time), we get:

The adults start at position 0 and move at 4 m/s. The kids start 4 meters ahead (at position 4) and move at 2 m/s.

Step 3 — Set up the system of equations

We want to find when they're at the same position, so we set the equations equal:

Step 4 — Solve for time

Subtract 2t from both sides:

Step 5 — Find how far the adults traveled

Substitute t = 2 back into the adults' equation:

Solution: Method 2 — Relative Speed Analysis

Step 1 — Identify the gap and closing rate

The kids have a 4-meter head start. The adults are gaining ground at a rate equal to the difference in their speeds:

Step 2 — Calculate time to close the gap

Using the relationship: time = distance ÷ rate

Step 3 — Find the adults' distance

In 2 seconds at 4 m/s, the adults travel:

System of Equations Summary

Verification

Let's check that both teams are at the same position after 2 seconds:

Both teams are indeed at the 8-meter mark after 2 seconds, confirming our solution.

Common Pitfalls

The Underlying Pattern

This is a classic "catch-up" problem that follows a general pattern. For any situation where a faster object chases a slower object with a head start:

They meet when the equations are equal, which gives:

This formula works for any catch-up scenario: pursuit problems in navigation, police chasing speeders, or even compound interest overtaking simple interest. The key insight is that the difference in rates determines how quickly the gap closes.

Real Applications

This mathematical structure appears frequently in real scenarios:

• Traffic and Navigation: A faster car leaving later to catch up to a slower car that left earlier — common in GPS route planning.
• Manufacturing: A faster production line trying to make up for lost time after starting behind schedule.
• Finance: Comparing investment growth rates where one investment starts with a higher principal but lower return rate.
• Project Management: Calculating when a team working overtime can catch up to a project that started ahead of schedule.

What If?