Corey and Trina both track the number of miles they bike each week. Corey was already biking 60 miles per week, and started biking 20 more miles each week. Trina was already biking 90 miles per week, and started biking 10 more miles each week. Is there a time when they bike the same number of miles each week?

Question

The relationships between x, the number of weeks, and y, the number of miles Corey and Trina bike each week, are graphed on the coordinate plane below. What is the point of intersection?
Corey
Trina
0
1
2
3
4
5
6
7
8
9
10
0
20
40
60
80
100
120
140
160
180
200
Time (weeks)
Number of miles
The point of intersection is (
,
).

Answer 1

Let's define the number of miles that Corey and Trina bike each week as functions of the number of weeks \(x\).

• Corey's miles: Corey starts at 60 miles and increases by 20 miles each week. Therefore, his weekly biking miles can be expressed as: \[ y_C = 60 + 20x \]

• Trina's miles: Trina starts at 90 miles and increases by 10 miles each week. Thus, her weekly biking miles can be expressed as: \[ y_T = 90 + 10x \]

We want to find the point at which Corey and Trina bike the same number of miles each week, meaning we need to set the two equations equal to each other: \[ 60 + 20x = 90 + 10x \]

Now, let's solve this equation for \(x\):

1. Subtract \(10x\) from both sides: \[ 60 + 10x = 90 \]
2. Subtract 60 from both sides: \[ 10x = 30 \]
3. Divide both sides by 10: \[ x = 3 \]

Now that we have \(x = 3\), we can use it to find the number of miles both Corey and Trina are biking after 3 weeks.

Using Corey’s equation: \[ y_C = 60 + 20(3) = 60 + 60 = 120 \]

Using Trina’s equation: \[ y_T = 90 + 10(3) = 90 + 30 = 120 \]

Both Corey and Trina bike 120 miles after 3 weeks.

Thus, the point of intersection is: \[ \text{(3, 120)} \]

So, the answer is: The point of intersection is (3, 120).