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Let's analyze the given questions related to direct proportion and direct variation.

1. **Which of the following equations represents a direct proportion?**
    - A. \( y = 3x + 1 \)
    - B. \( y = 1/x \)
    - C. \( y = 4x \)
    - D. \( y = x - 5 \)

   **Answer:** \( y = 4x \) (Option C) represents a direct proportion because it is of the form \( y = kx \) where \( k \) is a constant.

2. **Select the equation that shows a direct variation:**
    - A. \( y = x^2 \)
    - B. \( y = 2x \)
    - C. \( y = x/2 \)
    - D. \( y = x + 3 \)

   **Answer:** \( y = 2x \) (Option B) shows a direct variation because it is of the form \( y = kx \).

3. **Identify the equation that illustrates a direct proportionality:**
    - A. \( y = 5x + 3 \)
    - B. \( y = 3/x \)
    - C. \( y = 7x \)
    - D. \( y = x - 7 \)

   **Answer:** \( y = 7x \) (Option C) illustrates direct proportionality as it is of the form \( y = kx \).

4. **Which equation describes a direct relationship between \( x \) and \( y \)?**
    - A. \( y = x + 10 \)
    - B. \( y = 6x \)
    - C. \( y = 5/x \)
    - D. \( y = x - 2 \)

   **Answer:** \( y = 6x \) (Option B) describes a direct relationship because it is of the form \( y = kx \).

Do you want details or have any questions?

Here are five related questions to expand your understanding:

1. What is the difference between direct proportion and inverse proportion?
2. How do you identify a direct variation equation from a graph?
3. Can a direct proportion equation have a negative constant \( k \)?
4. What real-life examples can illustrate direct proportionality?
5. How would you convert an equation like \( y = 5x + 2 \) to show direct proportionality?

**Tip:** Always remember that in a direct proportion, as one variable increases, the other variable increases at a constant rate.

Determine which equations represent direct proportion, direct variation, and direct proportionality among the given options.

Understand how to identify direct proportion, direct variation, and direct proportionality in equations. This problem covers key concepts in algebra, including identifying relationships represented by equations. Ideal for students in Grades 7-9.

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