yler and Emma both track the number of miles they run each week. Tyler was already running 10 miles per week, and started running 2 more miles each week. Emma was already running 30 miles per week, and started running 3 less 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 Tyler and Emma run each week, are graphed on the coordinate plane below. What is the point of intersection?
Tyler
Emma
0
1
2
3
4
5
6
7
8
9
10
0
3
6
9
12
15
18
21
24
27
30
Time (weeks)
Number of miles
The point of intersection is (
,
).

Answer 1

To find the point of intersection of the number of miles Tyler and Emma run each week, we can set up equations for both of them based on the information given.

For Tyler:

• Starting miles: 10
• Increased by 2 miles each week

So, the equation for Tyler's miles after \(x\) weeks is: \[ y_T = 10 + 2x \]

For Emma:

• Starting miles: 30
• Decreased by 3 miles each week

So, the equation for Emma's miles after \(x\) weeks is: \[ y_E = 30 - 3x \]

Setting the equations equal to find the intersection: \[ 10 + 2x = 30 - 3x \]

Now, solve for \(x\):

1. Add \(3x\) to both sides: \[ 10 + 5x = 30 \]
2. Subtract 10 from both sides: \[ 5x = 20 \]
3. Divide by 5: \[ x = 4 \]

Now, substitute \(x = 4\) back into either equation to find \(y\). Using Tyler’s equation: \[ y_T = 10 + 2(4) = 10 + 8 = 18 \]

Using Emma’s equation for confirmation: \[ y_E = 30 - 3(4) = 30 - 12 = 18 \]

Thus, both equations give the same value for \(y\).

Therefore, the point of intersection, where Tyler and Emma run the same number of miles in 4 weeks, is: (4, 18).

The final answer is: The point of intersection is (4, 18).