Steven and Irene both track the number of miles they run each week. Steven was already running 20 miles per week, and started running 4 more miles each week. Irene was already running 30 miles per week, and started running 2 more miles each week. Is there a time when they run the same number of miles each week?

Question

The relationships between x, the number of weeks, and y, the number of miles Steven and Irene run each week, are graphed on the coordinate plane below. What is the point of intersection?
Steven
Irene
0
1
2
3
4
5
6
7
8
9
10
0
8
16
24
32
40
48
56
64
72
80
Time (weeks)
Number of miles

Answer 1

To determine if there is a time when Steven and Irene run the same number of miles each week, we can set up equations based on the information provided:

1. For Steven:

   • He starts at 20 miles per week and increases his mileage by 4 miles each week.
   • The equation for Steven's miles per week after \(x\) weeks is: \[ S(x) = 20 + 4x \]

2. For Irene:

   • She starts at 30 miles per week and increases her mileage by 2 miles each week.
   • The equation for Irene's miles per week after \(x\) weeks is: \[ I(x) = 30 + 2x \]

Next, we set these two equations equal to each other to find the number of weeks \(x\) when they run the same number of miles: \[ 20 + 4x = 30 + 2x \]

Now, we will solve for \(x\):

1. Subtract \(2x\) from both sides: \[ 20 + 2x = 30 \]

2. Subtract 20 from both sides: \[ 2x = 10 \]

3. Divide by 2: \[ x = 5 \]

Now that we have \(x = 5\), we can find the number of miles they both run at this time:

• For Steven: \[ S(5) = 20 + 4(5) = 20 + 20 = 40 \text{ miles} \]

• For Irene: \[ I(5) = 30 + 2(5) = 30 + 10 = 40 \text{ miles} \]

Thus, at \(x = 5\) weeks, both Steven and Irene are running 40 miles each week. Therefore, the point of intersection on the graph is (5, 40).

This confirms that there is indeed a time when they run the same number of miles, and that time is at 5 weeks with both running 40 miles.