Question

practice 2 : making an argument
date 12/19/23 period 5
you and a friend are going jogging at the springs preserve, but start at different locations. you start at the trailhead and jog at 10 miles per hour. your friend starts 4 miles from the trailhead and 8 miles per hour.
a. write a system of linear equations that represents this situation.
image of a map with you and your friend labels
b. now graph the system of linear equations that represents this situation.
chart with x and y axes, grid lines
c. your friend says that after an hour of biking, you will both be at the same location on the trail. is your friend correct? use the graph from part (a) to explain your answer. make sure to include your clear claim, use supporting evidence/details, a counterargument, and a final conclusion.

Explanation:

Part A

Step1: Define variables

Let \( x \) be the time in hours and \( y \) be the distance from the trailhead in miles.

Step2: Your equation

You start at the trailhead (\( y = 0 \) when \( x = 0 \)) and jog at 10 miles per hour. So the equation is \( y = 10x \).

Step3: Friend's equation

Your friend starts 4 miles from the trailhead (\( y = 4 \) when \( x = 0 \)) and jogs at 8 miles per hour. So the equation is \( y = 8x + 4 \).

Step4: System of equations

The system is \(

$$\begin{cases} y = 10x \\ y = 8x + 4 \end{cases}$$

\)

Step1: Graph your equation (\( y = 10x \))

• When \( x = 0 \), \( y = 0 \). Plot the point \( (0, 0) \).
• When \( x = 1 \), \( y = 10 \). Plot the point \( (1, 10) \).
• Draw a straight line through these points.

Step2: Graph friend's equation (\( y = 8x + 4 \))

• When \( x = 0 \), \( y = 4 \). Plot the point \( (0, 4) \).
• When \( x = 1 \), \( y = 8(1) + 4 = 12 \). Plot the point \( (1, 12) \).
• Draw a straight line through these points.

Step1: Evaluate at \( x = 1 \)

• For your equation (\( y = 10x \)): When \( x = 1 \), \( y = 10(1) = 10 \).
• For friend's equation (\( y = 8x + 4 \)): When \( x = 1 \), \( y = 8(1) + 4 = 12 \).

Step2: Compare distances

At \( x = 1 \) (1 hour), your distance is 10 miles and your friend's distance is 12 miles. These are not equal.

Step3: Graphical check

On the graph, the two lines (representing your and your friend's distance - time relationships) do not intersect at \( x = 1 \). The intersection point (where you meet) can be found by solving \( 10x = 8x + 4 \), which gives \( 2x = 4 \) or \( x = 2 \). So you meet at \( x = 2 \) hours, not \( x = 1 \) hour.

Answer:

The system of linear equations is \( \boldsymbol{

$$\begin{cases} y = 10x \\ y = 8x + 4 \end{cases}$$

} \)

Part B