Use the image to answer the question.

Question

Four lines are graphed on a coordinate plane. The x-axis is labeled Hours and ranges from 0 to 2.5 in increments of 0.5. The y-axis is labeled Miles and ranges from 0 to 80 in increments of 10.

Sara went to meet her friend Rianna in Rome, but Rianna left Rome on a train 15 minutes earlier. The distance traveled by the train, in miles, with respect to time can be modeled by the equation y=40x+10
. Sara can buy a ticket on a faster train that will leave in 10 minutes and follow the same route. If the distance of the faster train with respect to time can be modeled by the equation y=60x−10
, which two lines on the graph show the solution to when Sara’s train catches up to Rianna’s train?

(1 point)
Responses

Line 2 and Line 3
Line 2 and Line 3

Line 1 and Line 2
Line 1 and Line 2

Line 3 and Line 4
Line 3 and Line 4

Line 1 and Line 4

Answer 1

To determine which two lines on the graph represent when Sara's train catches up to Rianna's train, we need to analyze the equations for the distance traveled by both trains.

1. Rianna's train: The equation is \( y = 40x + 10 \).

   • This represents the distance traveled by Rianna as a function of time \( x \) (in hours).

2. Sara's train: The equation is \( y = 60x - 10 \).

   • This represents the distance traveled by Sara as a function of time \( x \), but keep in mind that Sara starts her journey 10 minutes (or \( \frac{1}{6} \) hour) after Rianna.

To find the point at which Sara's train catches up to Rianna's train, we set the two equations equal to each other:

\[ 40x + 10 = 60x - 10 \]

Now, simplify the equation:

\[ 10 + 10 = 60x - 40x \] \[ 20 = 20x \] \[ x = 1 \]

This means that after 1 hour, Sara's train catches up to Rianna's train.

To find the corresponding \( y \)-value (distance in miles) when Sara catches up:

Substituting \( x = 1 \) into either equation:

For Rianna's equation: \[ y = 40(1) + 10 = 50 \]

For Sara's equation: \[ y = 60(1) - 10 = 50 \]

Both results indicate that at \( x = 1 \) hour, they are both at 50 miles.

Now we can identify which lines correspond to these equations:

• Line 1 corresponds to \( y = 40x + 10 \) (Rianna's train).
• Line 2 corresponds to \( y = 60x - 10 \) (Sara's train).

Therefore, we can conclude that the lines representing when Sara’s train catches up to Rianna’s train are:

Line 1 and Line 2.

So the correct answer is Line 1 and Line 2.