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Brandi and her daughter Ella, are training for a hiking challenge. Because Brandi hikes at a slower pace than her daughter, she begins the practice hike two hours earlier. If Brandi averages a pace of 4 mph, the linear equation y=4x

can be used to model her distance, y, in miles with respect to her time, x, in hours. If Ella averages a pace of 6 mph, and begins two hours after her mom, the linear equation y=6x−12
can be used to model her distance, y, in miles with respect to time, x, in hours. The graph of which two lines can be used to find the time and distance when Ella catches up with her mother? After how many hours and miles (estimated) does Ella catch up?

(1 point)
Line
and Line
can be used to determine when Ella catches up to her mother.

(Line numbers in your answer above should be numerically in order from least to greatest)

Ella catches up in
hours and has traveled
​​​​​​​ miles.

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1 answer

To determine when Ella catches up with her mother, we need to analyze the equations given:

  1. Brandi's distance equation: \( y = 4x \)
  2. Ella's distance equation: \( y = 6x - 12 \)

To find when Ella catches up with Brandi, we set the two equations equal to each other:

\[ 4x = 6x - 12 \]

Now, we solve for \( x \):

\[ 4x - 6x = -12 \] \[ -2x = -12 \] \[ x = 6 \]

This means Ella catches up with her mother 6 hours after Brandi starts her hike. Since Brandi starts 2 hours earlier, we can also conclude that Ella is hiking for \( 6 - 2 = 4 \) hours.

Next, to find the distance they have traveled when they meet, we can substitute \( x \) back into either equation. Using Brandi's equation:

\[ y = 4(6) = 24 , \text{miles} \]

or using Ella's equation (to verify):

\[ y = 6(4) = 24 , \text{miles} , \text{(as Ella hiked 4 hours)} \]

Thus, we can summarize:

  • Line 1: \( y = 4x \) (Brandi)
  • Line 2: \( y = 6x - 12 \) (Ella)

Ella catches up in 6 hours and has traveled 24 miles.

So the answers are:

Line 1 and Line 2 can be used to determine when Ella catches up to her mother.

Ella catches up in 6 hours and has traveled 24 miles.