Cameron and Samara both track the number of miles they run each week. Cameron was already running 20 miles per week, and started running 4 more miles each week. Samara 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 Cameron and Samara run each week, are graphed on the coordinate plane below. What is the point of intersection?
Cameron
Samara
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
The point of intersection is (
,
).

Answer 1

To find the point of intersection for the miles Cameron and Samara run each week, we can first establish the equations for the miles run by both individuals as a function of the number of weeks \( x \).

Cameron:

• Initial miles: 20 miles/week
• Additional miles per week: 4 miles/week

The equation for Cameron's miles run each week is: \[ y_{\text{Cameron}} = 20 + 4x \]

Samara:

• Initial miles: 30 miles/week
• Additional miles per week: 2 miles/week

The equation for Samara's miles run each week is: \[ y_{\text{Samara}} = 30 + 2x \]

To find the point of intersection, we need to set the two equations equal to each other to find \( x \):

\[ 20 + 4x = 30 + 2x \]

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

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

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

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

Now we can find the number of miles run by either Cameron or Samara at \( x = 5 \):

Using Cameron's equation: \[ y_{\text{Cameron}} = 20 + 4(5) = 20 + 20 = 40 \]

Using Samara's equation: \[ y_{\text{Samara}} = 30 + 2(5) = 30 + 10 = 40 \]

Both Cameron and Samara run 40 miles when \( x = 5 \).

Thus, the point of intersection is: \[ (5, 40) \]

Therefore, the answer is: The point of intersection is \( (5, 40) \).