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 solve this, let's examine the data and find the linear relationship. A linear equation takes the form:

\[
y = mx + b
\]

where:
- \( m \) is the slope (the rate of change of elevation per minute),
- \( b \) is the y-intercept (the initial elevation when \( x = 0 \)).

### Step 1: Calculate the slope (m)
The slope \( m \) is the change in \( y \) divided by the change in \( x \). We can use any two points from the table to find it. Let’s use the points (1, 180) and (2, 160).

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

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

### Step 2: Find the y-intercept (b)
To find \( b \), we can substitute any point from the table into the equation \( y = mx + b \). Let’s use the point (1, 180).

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

\[
180 = -20 + b
\]

\[
b = 200
\]

So, the y-intercept is \( b = 200 \).

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

\[
y = -20x + 200
\]

### Answer: 
The correct equation is **D. \( y = -20x + 200 \)**.

Would you like more details on the steps or have any questions?

Here are 5 related questions to deepen your understanding:
1. How would the equation change if the hiker ascended the hill instead?
2. What would the y-intercept be if the hiker started at 220 feet?
3. Can you graph this equation based on the table data?
4. How would you interpret the slope if the units were kilometers instead of feet?
5. What happens to the slope if the time intervals between measurements are not uniform?

**Tip:** The slope in a linear equation represents the rate of change. Negative slope means a decrease, while positive slope means an increase.

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 representing a hiker's descent using the slope-intercept form. The problem uses data points on time and elevation to calculate the slope and y-intercept, with a solution suitable for Grades 6-8 students. Includes step-by-step explanation and the final equation y = -20x + 200.

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