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Ann begins her practice hike 2 hours earlier than Ella If Ann averages her 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 6mph and begins 2 hours after Ann the linear equation y= 6x -12 can be used which two lines on the graph can be used to find the time and distwhen Ella catches up with Ann

Line 1 (0,4) goes across and increases points it crosses line 2 at 2,8?), line 3 at (4,12?)and line 4 at (6,16?)
Line 2 (0,0) crosses line 1at (2,8?) line 3at (6,24?) line 4 at (8,32?)
Line 3 (2,0) crosses line 1at (4,12?) and line 2 at (6,24?) does not cross line 4
Line 4 (4,0) crosses line 1 at (6,16?) and line 2 at (8,32?)

1 answer

To find the time and distance when Ella catches up with Ann, we need to determine where the two lines representing their distances cross.

  1. Ann's Distance Equation:

    • \( y = 4x \)
    • This represents Ann's distance in miles after \( x \) hours, with a speed of 4 mph.

  2. Ella's Distance Equation:

    • \( y = 6x - 12 \)
    • This represents Ella's distance in miles after \( x \) hours, starting 2 hours later than Ann, with a speed of 6 mph.

To find the point where Ella catches up with Ann, we set their equations equal to each other:

\[ 4x = 6x - 12 \]

Now, solving for \( x \):

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

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

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

So, at \( x = 6 \) hours, both Ann and Ella will have covered a distance of \( 24 \) miles.

Now let's look at which lines on the graph can be used to find this point:

  • Ella starts 2 hours later than Ann, so she catches up at \( x = 6 \) hours.
  • The corresponding distance when both have gone the same distance is \( y = 24 \) miles.

Next, let’s evaluate the lines provided based on given intersection points.

  • Line 1 intersects with Line 2 at (2, 8), with Line 3 at (4, 12), and with Line 4 at (6, 16).
  • Line 2 intersects with Line 1 at (2, 8), with Line 3 at (6, 24), and with Line 4 at (8, 32).
  • Line 3 intersects with Line 1 at (4, 12), with Line 2 at (6, 24), but does not intersect Line 4.
  • Line 4 intersects with Line 1 at (6, 16) and with Line 2 at (8, 32).

The correct lines to identify when Ella catches up with Ann are:

Line 2 (0, 0) and Line 3 (2, 0):

  • Line 2 intersects Line 3 at (6, 24).

Thus, the lines that will help you find when Ella catches up with Ann are Line 2 and Line 3, where the intersection occurs at the point (6, 24).