Olivia and Addison both track the number of miles they bike each week. Olivia was already biking 60 miles per week, and started biking 20 more miles each week. Addison 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 Olivia and Addison bike each week, are graphed on the coordinate plane below. What is the point of intersection?
Olivia
Addison
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

To find when Olivia and Addison bike the same number of miles, let's set up equations for the number of miles they bike per week:

1. Olivia's miles per week:

   • Initial: 60 miles
   • Increase: 20 miles each week
   • After \( x \) weeks, Olivia's miles: \( y = 60 + 20x \)

2. Addison's miles per week:

   • Initial: 90 miles
   • Increase: 10 miles each week
   • After \( x \) weeks, Addison's miles: \( y = 90 + 10x \)

We set the two equations equal to find \( x \) when they bike the same distance:

\[ 60 + 20x = 90 + 10x \]

Now solve for \( x \):

1. Subtract \( 10x \) from both sides: \[ 60 + 10x = 90 \]

2. Subtract 60 from both sides: \[ 10x = 30 \]

3. Divide by 10: \[ x = 3 \]

Now, substitute \( x = 3 \) back into either equation to find \( y \) (the number of miles):

Using Olivia's equation: \[ y = 60 + 20(3) = 60 + 60 = 120 \]

Therefore, the number of miles both Olivia and Addison bike at the same time is 120 miles.

Therefore, the point of intersection is:

\[ (3, 120) \]

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