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 tasked with finding a linear equation that represents the relationship between the time (in minutes, \( x \)) and the elevation (in feet, \( y \)).

We know that the equation of a line can be written in the slope-intercept form:

\[
y = mx + b
\]

where:
- \( m \) is the slope of the line (rate of change),
- \( b \) is the y-intercept (the value of \( y \) when \( x = 0 \)).

### Step 1: Find the slope (\( m \))
The slope \( m \) is the rate of change in \( y \) with respect to \( x \), which can be found using the formula:

\[
m = \frac{\Delta y}{\Delta x}
\]

Let's take two points from the table:
- (1, 180)
- (2, 160)

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

The slope \( m = -20 \), meaning the hiker descends 20 feet per minute.

### Step 2: Find the y-intercept (\( b \))
We can substitute one of the points from the table and the value of \( m \) into the equation \( y = mx + b \) to solve for \( b \).

Using the point (1, 180):
\[
180 = -20(1) + b
\]
\[
180 = -20 + b
\]
\[
b = 200
\]

### Step 3: Write the equation
Now that we have \( m = -20 \) and \( b = 200 \), the equation is:

\[
y = -20x + 200
\]

Thus, the linear equation that represents the relationship between the time and the elevation is:

\[
y = -20x + 200
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How would the equation change if the hiker was ascending instead of descending?
2. How can you verify the slope using other pairs of points from the table?
3. What does the y-intercept represent in the context of this problem?
4. How would the equation look if the rate of descent doubled?
5. How can you graph this equation and interpret its features?

**Tip:** When finding the slope, always check at least two pairs of points to ensure consistency.

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 find the linear equation that represents a hiker's descent down a hill based on the time and elevation data provided. This example shows how to calculate the slope and y-intercept to form the equation y = -20x + 200. Suitable for students in Grades 6-8.

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