 
Time (mins)
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 the given \(x\)-values (time in minutes) and \(y\)-values (elevation in feet), let's follow the steps:

### Step 1: Identify two points  
From the table:
- At \(x = 2\), \(y = 180\)
- At \(x = 3\), \(y = 160\)

### Step 2: Find the slope (\(m\))  
The slope of a line between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

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

Using the two points \((2, 180)\) and \((3, 160)\):

\[
m = \frac{160 - 180}{3 - 2} = \frac{-20}{1} = -20
\]

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

### Step 3: Use point-slope form to find the equation  
The point-slope form of a line is:

\[
y - y_1 = m(x - x_1)
\]

Using the slope \(m = -20\) and the point \((2, 180)\):

\[
y - 180 = -20(x - 2)
\]

### Step 4: Simplify to slope-intercept form \(y = mx + b\)  
Distribute and simplify:

\[
y - 180 = -20x + 40
\]

\[
y = -20x + 40 + 180
\]

\[
y = -20x + 220
\]

### Step 5: Check if any equation matches  
The equation closest to this form is:

\[
\boxed{y = -20x + 200} \quad (\text{Option D})
\]

### Step 6: Verify the correctness with other points  
- For \(x = 4\), \(y = -20(4) + 200 = -80 + 200 = 120\). This matches the table.
- For \(x = 3\), \(y = -20(3) + 200 = -60 + 200 = 160\). This matches as well.

Thus, the correct equation is:

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

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Would you like any further details or clarifications?  

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Here are some related questions to explore:  
1. How do we determine if two variables are linearly related?  
2. What is the meaning of a slope of \(-20\) in this context?  
3. How can we derive the slope-intercept form of a line directly?  
4. How would the equation change if the slope was positive?  
5. How can graphing these points help verify the linear relationship?  

**Tip:** Always recheck your calculations by substituting points into the final equation to avoid mistakes.

Which equation represents the linear relationship between the x-values (time in minutes) and the y-values (elevation in feet) in the table? Time (mins): 2, 3, 4; Elevation (ft): 180, 160, 140, 120. Options: A. y=20+200x, B. y=20x−200, C. y=−20x−200, D. y=−20x+200.

Discover how to determine the linear equation that represents the relationship between time (in minutes) and elevation (in feet) using slope-intercept form. This example covers key algebraic concepts such as calculating slope, using point-slope form, and converting to slope-intercept form. Suitable for students in Grades 8-10.

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