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Four lines are graphed on a coordinate plane. The x-axis is labeled Hours and ranges from 0 to 9 in increments of 1. The y-axis is labeled Miles and ranges from 0 to 40 in increments of 10.

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?

Option #1: Line 1 and Line 2

Option #2: Line 2 and Line 3

Option #3: Line 3 and Line 4

Option #4: Line 1 and Line 4

(1 point)
The lines in option #
can be used to determine when Ella catches up

1 answer

To find out when Ella catches up with Brandi, you need to look for the point where their distances, represented by their respective equations, are equal.

The equations are:

  • Brandi's distance: \( y = 4x \)
  • Ella's distance: \( y = 6x - 12 \)

To find the point where these two lines intersect, you would set them equal to each other:

\[ 4x = 6x - 12 \]

Solving for \( x \): \[ 12 = 6x - 4x \] \[ 12 = 2x \] \[ x = 6 \]

Now, substitute \( x = 6 \) back into either equation to find \( y \):

Using Brandi's equation: \[ y = 4(6) = 24 \]

So at \( x = 6 \) hours, both Brandi and Ella are at \( y = 24 \) miles. Therefore, we need the corresponding lines that represent these equations on the graph in order to find where these values intersect.

From the information given, it looks like:

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

Thus, the correct option that includes the lines for Brandi and Ella (where Ella catches up to Brandi) is:

Option #1: Line 1 and Line 2

So, the lines in option #1 can be used to determine when Ella catches up.