Question
Jordan will hike the trail shown at a rate of 3 mi/h. Write a linear equation to represent the distance Jordan still has to walk after x hours. What does the y-intercept of the equation represent?
Answers
GPT-4o mini
To write a linear equation representing the distance Jordan still has to walk after \( x \) hours, we start by defining the variables involved:
- Let \( d \) represent the distance remaining to walk (in miles).
- Let \( x \) represent the time spent hiking (in hours).
Since Jordan is hiking the trail at a rate of 3 mi/h, the distance walked after \( x \) hours can be represented as \( 3x \).
If \( D \) is the total distance of the trail (in miles), then the remaining distance \( d \) after \( x \) hours can be expressed as:
\[
d = D - 3x
\]
This can be rearranged to put it in the form of a linear equation:
\[
d = -3x + D
\]
In this equation, the slope is \(-3\) (indicating that distance decreases as time increases) and \( D \) is the y-intercept.
### What does the y-intercept represent?
The y-intercept \( D \) represents the initial distance that Jordan has to walk before starting his hike (i.e., the total distance of the trail). When \( x = 0 \) (which means no time has passed), the distance remaining to walk is equal to the total distance of the trail, \( D \). Thus, the y-intercept gives us the total distance of the trail before Jordan begins hiking.
- Let \( d \) represent the distance remaining to walk (in miles).
- Let \( x \) represent the time spent hiking (in hours).
Since Jordan is hiking the trail at a rate of 3 mi/h, the distance walked after \( x \) hours can be represented as \( 3x \).
If \( D \) is the total distance of the trail (in miles), then the remaining distance \( d \) after \( x \) hours can be expressed as:
\[
d = D - 3x
\]
This can be rearranged to put it in the form of a linear equation:
\[
d = -3x + D
\]
In this equation, the slope is \(-3\) (indicating that distance decreases as time increases) and \( D \) is the y-intercept.
### What does the y-intercept represent?
The y-intercept \( D \) represents the initial distance that Jordan has to walk before starting his hike (i.e., the total distance of the trail). When \( x = 0 \) (which means no time has passed), the distance remaining to walk is equal to the total distance of the trail, \( D \). Thus, the y-intercept gives us the total distance of the trail before Jordan begins hiking.