A hiker is descending a hill. The table represents the hiker's elevation in feet, y
, after x
 minutes.
 
Time (mins)	1
2
3
4
Elevation (ft)	180
160
140
120

Which equation represents the linear relationship between the x
-values and the y
-values in the table?
A.y=20+200x

B.y=20x−200

C.y=−20x−200

D.y=−20x+200


To determine which equation represents the linear relationship between the time \( x \) (in minutes) and the elevation \( y \) (in feet), we can follow these steps:

1. **Check for a constant rate of change (slope):**  
   The hiker's elevation decreases by the same amount for each minute. Let's calculate the change in \( y \) (elevation) for each minute:
   
   \[
   \text{Change in } y = 160 - 180 = -20 \quad (\text{from minute 1 to minute 2}),
   \]
   \[
   \text{Change in } y = 140 - 160 = -20 \quad (\text{from minute 2 to minute 3}),
   \]
   \[
   \text{Change in } y = 120 - 140 = -20 \quad (\text{from minute 3 to minute 4}).
   \]
   
   The rate of change (slope) is \( -20 \) feet per minute, indicating the hiker descends 20 feet each minute.

2. **Identify the slope-intercept form:**
   The general form of a linear equation is \( y = mx + b \), where:
   - \( m \) is the slope (which we've determined to be \( -20 \)).
   - \( b \) is the y-intercept, the value of \( y \) when \( x = 0 \).

3. **Determine the y-intercept:**
   When \( x = 0 \), the hiker has not yet started descending. The hiker starts at an elevation of 200 feet. Therefore, \( b = 200 \).

4. **Write the equation:**
   Using the slope \( m = -20 \) and y-intercept \( b = 200 \), the equation becomes:
   \[
   y = -20x + 200
   \]

Thus, the correct equation is:

**D. \( y = -20x + 200 \)**

---

Would you like further details on how we derived the slope or intercept?

Here are 5 related questions to extend your understanding:
1. How do you find the slope of a line from two points?
2. What is the significance of the y-intercept in a linear equation?
3. How can you convert an equation into slope-intercept form?
4. How do you interpret the slope in a real-world context like this problem?
5. How can you check if a set of points lie on the same line?

**Tip:** Always check if the change in y-values is consistent, as this confirms the relationship is linear.

A hiker is descending a hill. The table represents the hiker's elevation in feet, y, after x minutes. Time (mins): 1, 2, 3, 4. Elevation (ft): 180, 160, 140, 120. Which equation represents the linear relationship between the x-values and the y-values in the table? A) y = 20 + 200x, B) y = 20x − 200, C) y = −20x − 200, D) y = −20x + 200

Learn how to find the linear equation representing a hiker's descent over time. Using the slope-intercept form and calculating the rate of change, this problem explains how the hiker's elevation decreases by 20 feet per minute, with a starting height of 200 feet. Suitable for Grades 8-10.

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