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?

We are asked to find the equation representing the linear relationship between the time \( x \) (in minutes) and the elevation \( y \) (in feet).

### Step 1: Identify the form of the linear equation.
A linear relationship is typically expressed in the form:

\[
y = mx + b
\]

Where:
- \( m \) is the slope (rate of change of \( y \) with respect to \( x \))
- \( b \) is the y-intercept (the elevation when \( x = 0 \))

### Step 2: Calculate the slope \( m \).
The slope is the change in \( y \) (elevation) per unit change in \( x \) (time). Using two points from the table, we can calculate it:

Points: \( (1, 180) \) and \( (2, 160) \).

The formula for the slope is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points:

\[
m = \frac{160 - 180}{2 - 1} = \frac{-20}{1} = -20
\]

So, the slope \( m = -20 \).

### Step 3: Find the y-intercept \( b \).
Using the slope \( m = -20 \), we can plug in one of the points to find \( b \). Let's use \( (1, 180) \):

\[
y = mx + b
\]
\[
180 = -20(1) + b
\]
\[
180 = -20 + b
\]
\[
b = 180 + 20 = 200
\]

### Step 4: Write the equation.
Now that we have the slope \( m = -20 \) and the y-intercept \( b = 200 \), the equation representing the linear relationship is:

\[
y = -20x + 200
\]

### Final Answer:
The equation is:

\[
y = -20x + 200
\]

Would you like further clarification or details?

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Here are 5 related questions to deepen understanding:
1. What does the slope of the equation tell you about the hiker's descent?
2. How would the equation change if the hiker descended at a faster rate?
3. What would the graph of this linear equation look like?
4. How can you interpret the y-intercept in the context of this problem?
5. How would the equation be different if the hiker were ascending instead?

**Tip:** In a linear equation, the slope represents the rate of change, and it can help to understand how quickly one variable changes with respect to another.

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?

Learn how to determine the linear equation representing a hiker's descent over time. Using the data table provided, we calculate the slope and y-intercept to find the equation y = -20x + 200, which describes the relationship between time and elevation. Suitable for students in Grades 6-8.

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