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


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

From the table:

| Time \(x\) (mins) | Elevation \(y\) (ft) |
|-------------------|----------------------|
| 1                 | 180                  |
| 2                 | 160                  |
| 3                 | 140                  |
| 4                 | 120                  |

### Step 1: Identify the slope (rate of change)
Since the relationship is linear, the **slope** \(m\) of the line is the rate of change in the elevation per minute. Use the slope formula:

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

Pick two points from the table, say \((x_1, y_1) = (1, 180)\) and \((x_2, y_2) = (2, 160)\):

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

The slope \(m = -20\). This makes sense because the hiker is descending (losing elevation), so the slope is negative.

### Step 2: Use slope-intercept form of a line
The slope-intercept form is:

\[
y = mx + b
\]

where:
- \(m\) is the slope
- \(b\) is the y-intercept (the elevation at \(x = 0\))

We know \(m = -20\). Now we need to find \(b\).

### Step 3: Solve for the y-intercept \(b\)
Pick a point from the table, such as \((x, y) = (1, 180)\). Plug it into the equation \(y = -20x + b\):

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

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

### Step 4: Write the equation
The equation of the line is:

\[
y = -20x + 200
\]

### Step 5: Identify the correct answer
The correct answer is:

\[
\boxed{D. \, y = -20x + 200}
\]

---

Do you need further details or clarification?  

Here are 5 related questions to deepen your understanding:
1. How do you interpret the slope in the context of this problem?
2. What would the graph of this equation look like?
3. How does the y-intercept relate to the starting elevation of the hiker?
4. What happens to the elevation after 10 minutes?
5. How would the equation change if the hiker were ascending instead?

**Tip:** When working with tables, always check if the changes in \(x\) and \(y\) are consistent to confirm it's a linear relationship.

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 for a hiker descending a hill using the slope-intercept form. The problem involves analyzing time and elevation data, calculating the slope, and determining the y-intercept. Suitable for students in Grades 6-8.

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