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.
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Rianna's train: The equation is \( y = 40x + 10 \).
- This represents the distance traveled by Rianna as a function of time \( x \) (in hours).
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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.