Question

harold and troy got in trouble at football practice and have to run laps as a consequence. harold, who runs at a rate of 1 lap per minute, had completed 6 laps already when he was joined on the track by troy. troys pace is 3 laps per minute. at some point, the two will have run the same distance. how long will that take?
write a system of equations, graph them, and type the solution.
(graph: y - axis labeled laps from 0 to 10, x - axis labeled minutes from 0 to 10, grid lines, buttons for harold and troy at the bottom)

Explanation:

Step1: Define variables and write Harold's equation

Let \( x \) be the time in minutes after Troy starts running, and \( y \) be the number of laps run. Harold has already run 6 laps and runs at 1 lap per minute, so his equation is \( y = x + 6 \).

Step2: Write Troy's equation

Troy runs at 3 laps per minute, so his equation is \( y = 3x \).

Step3: Solve the system of equations

Set the two equations equal to each other: \( 3x = x + 6 \). Subtract \( x \) from both sides: \( 2x = 6 \). Divide both sides by 2: \( x = 3 \).

To graph: For \( y = x + 6 \), the y-intercept is 6, and the slope is 1. For \( y = 3x \), the y-intercept is 0, and the slope is 3. The lines intersect at \( x = 3 \), \( y = 9 \).

Answer:

It will take 3 minutes. The system of equations is \(

$$\begin{cases} y = x + 6 \\ y = 3x \end{cases}$$

\), and the solution is \( x = 3 \) (minutes) when they have run the same distance.