https://r2.erweima.ai/mathbot/image/17268499383541vhzjlbw.jpg

The image presents a problem about the relationship between price (\(p\)) and quantity (\(q\)) of an article, represented by a linear graph. The task is to determine and interpret the slope (or "pendiente") of the line.

### Key points from the image:
1. **Points on the line:**
   - The line passes through the point \((2, 4)\), where \(q = 2\) (in thousands) and \(p = 4\) (dollars).
   - The line also passes through the point \((8, 1)\), where \(q = 8\) (in thousands) and \(p = 1\) (dollars).

### Finding the slope:
The slope \(m\) of a line is calculated as the change in \(y\) (vertical change) divided by the change in \(x\) (horizontal change):

\[
m = \frac{\Delta p}{\Delta q} = \frac{p_2 - p_1}{q_2 - q_1}
\]

Where:
- \( (q_1, p_1) = (2, 4) \)
- \( (q_2, p_2) = (8, 1) \)

Substituting these values:

\[
m = \frac{1 - 4}{8 - 2} = \frac{-3}{6} = -\frac{1}{2}
\]

### Interpretation of the slope:
- The slope of \(-\frac{1}{2}\) means that for every increase of 1 unit in \(q\) (thousands of items), the price \(p\) decreases by \(\frac{1}{2}\) (or 0.5 dollars).
- This negative slope indicates an inverse relationship between price and quantity: as more items are purchased, the price per item decreases.

### Summary:
- **Slope**: \(-\frac{1}{2}\)
- **Interpretation**: For each additional 1,000 units bought, the price decreases by 0.5 dollars.

Would you like further clarification or details?

### Related Questions:
1. How does the slope affect the relationship between price and quantity?
2. What is the significance of a negative slope in economic models?
3. How would the line look if the relationship were directly proportional?
4. Could we model this relationship using an equation? What would it look like?
5. What real-world factors might influence the price-quantity relationship?

**Tip**: In economics, understanding the slope in demand and supply curves is crucial as it reflects sensitivity to price changes (elasticity).

Determine the slope and interpret the relationship between price and quantity from the linear graph showing points (2, 4) and (8, 1).

Learn how to calculate and interpret the slope of a linear graph representing the relationship between price and quantity of an article, using key points (2, 4) and (8, 1). Suitable for high school students studying linear equations.

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