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 find the equation that represents the linear relationship between time (\(x\)) and elevation (\(y\)) in the table, follow these steps:

### Step 1: Identify the slope (rate of change)
The slope of a line can be calculated as the change in \(y\) divided by the change in \(x\), also known as "rise over run." Let's calculate it using two points from the table:

- Point 1: \((1, 180)\)
- Point 2: \((2, 160)\)

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

Thus, the slope is \(-20\).

### Step 2: Use the slope-intercept form of a line
The slope-intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Since the slope is \(-20\), the equation is of the form:
\[
y = -20x + b
\]

### Step 3: Solve for the y-intercept (\(b\))
To find \(b\), plug in one of the points from the table. Let’s use the point \((1, 180)\):

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

Thus, the y-intercept is \(200\).

### Final Equation
The equation representing the linear relationship is:
\[
y = -20x + 200
\]

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

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Do you want more details or have any questions?

Here are 5 related questions:
1. How do you calculate the slope between two points?
2. What is the slope-intercept form of a line?
3. How do you determine the y-intercept from a given point and slope?
4. Can you graph a linear equation from its slope and y-intercept?
5. How does the slope affect the direction of a line?

**Tip**: The sign of the slope tells you the direction of the line: negative slopes indicate a descending line, while positive slopes indicate an ascending one.

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

Explore how to find the linear equation that represents a hiker's descent from elevation data over time. Using the slope-intercept form and slope calculation, this problem demonstrates key algebra concepts for middle school students. Perfect for Grades 6-8.

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