(c) What is the graph of the equation that represents the relationship between​ driven, D, and gas​ used, G?
A.
10
0
20
350
0
700
Gas (gal)
Distance (mi)



A coordinate system has a horizontal axis Gas (gallons) labeled from 0 to 20 in increments of 2 and a vertical axis Distance (miles) labeled from 0 to 700 in increments of 70. A line starts at the origin (0, 0) and rises from left to right passing through (12, 432).
B.
10
0
20
350
0
700
Gas (gal)
Distance (mi)



A coordinate system has a horizontal axis Gas (gallons) labeled from 0 to 20 in increments of 2 and a vertical axis Distance (miles) labeled from 0 to 700 in increments of 70. A line starts at the origin (0, 0) and rises from left to right passing through the approximate point (1, 280).
C.
10
0
20
350
0
700
Gas (gal)
Distance (mi)



A coordinate system has a horizontal axis Gas (gallons) labeled from 0 to 20 in increments of 2 and a vertical axis Distance (miles) labeled from 0 to 700 in increments of 70. A line starts at the origin (0, 0) and rises from left to right passing through (20, 360).
D.
10
0
20
350
0
700
Gas (gal)
Distance (mi)

To determine the correct graph of the equation representing the relationship between the **distance driven (D)** and **gas used (G)**, let's follow these steps:  

---

### **1. Identify the Linear Relationship:**

Since all graphs show a **line starting at the origin (0, 0)**, it suggests that the equation has the form:  
\[
D = mG
\]
where:
- \(D\) is the distance driven in miles,
- \(G\) is the gas used in gallons,
- \(m\) is the slope, representing **miles per gallon (mpg)**.

The slope \(m\) can be calculated as:
\[
m = \frac{\text{Change in Distance}}{\text{Change in Gas}}
\]

---

### **2. Examine Points in the Graphs:**

- **Option A:** A line passes through \( (12, 432) \)  
    - Slope: \( m = \frac{432 - 0}{12 - 0} = \frac{432}{12} = 36 \, \text{mpg} \)

- **Option B:** A line passes through \( (1, 280) \)  
    - Slope: \( m = \frac{280 - 0}{1 - 0} = 280 \, \text{mpg} \)  
    - **Unrealistic** because 280 mpg is extremely high.

- **Option C:** A line passes through \( (20, 360) \)  
    - Slope: \( m = \frac{360 - 0}{20 - 0} = \frac{360}{20} = 18 \, \text{mpg} \)

---

### **3. Conclusion:**
- **Option A** shows a realistic slope of **36 mpg**, and it matches the gradual rise visible in the given description.
- **Option B** suggests a very steep slope of 280 mpg, which is not reasonable for most vehicles.
- **Option C** indicates a slope of 18 mpg, but this would require a flatter graph.

Thus, **Option A** is the best representation of the relationship between distance driven and gas used.

---

### Do you want further details or have any questions?  
Here are five related questions:  
1. What is the general formula for linear relationships?  
2. How do you interpret the slope in real-world contexts like miles per gallon?  
3. How would you find the equation of the line given two points?  
4. What are the units of the slope in this case?  
5. How can you determine if a line passes through a particular point?  

**Tip:** When working with linear relationships, always check the slope to ensure it matches the real-world context (e.g., reasonable mpg values).

What is the graph of the equation that represents the relationship between driven, D, and gas used, G? Options include A, B, C, and D with corresponding graphs and slope values.

Explore how to graph the relationship between the distance driven (D) and gas used (G) using linear equations. Understand the concept of slope in real-world applications like miles per gallon. This solution includes a detailed analysis of slope values and graph interpretations suitable for Grades 8-10.

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