https://file.aiquickdraw.com/mathbot/file/1726796220939y6awh466.jpg

Let's break down each of the questions based on the values provided on the cards. Here's how we can solve each question:

### 1. For card A, find the constant of proportionality.
Card A gives:  
\( x = 2 \), \( y = 11 \).

In a direct variation, \( y = kx \), where \( k \) is the constant of proportionality. Solving for \( k \):

\[
k = \frac{y}{x} = \frac{11}{2} = 5.5
\]

So, the constant of proportionality for card A is **5.5**.

### 2. For card E, find the value of "k."
Card E gives:  
\( x = -10 \), \( y = 2 \).

Again, using \( k = \frac{y}{x} \):

\[
k = \frac{2}{-10} = -0.2
\]

The value of \( k \) for card E is **-0.2**.

### 3. Which card shows a relationship where the constant of proportionality is 64?
We need to check which card has \( k = 64 \). For this, we apply \( k = \frac{y}{x} \) for each card:

- For Card B: \( \frac{y}{x} = \frac{5}{15} = \frac{1}{3} \)
- For Card C: \( \frac{y}{x} = \frac{16}{\frac{1}{4}} = 16 \times 4 = 64 \)

Thus, **Card C** shows a relationship where the constant of proportionality is **64**.

### 4. Write a direct variation equation to represent the relationship in card D.
Card D gives:  
\( x = 3 \), \( y = \frac{3}{7} \).

Using \( y = kx \), solve for \( k \):

\[
k = \frac{y}{x} = \frac{\frac{3}{7}}{3} = \frac{1}{7}
\]

Thus, the direct variation equation for card D is:

\[
y = \frac{1}{7}x
\]

### 5. Write a direct variation equation to represent the relationship in card F.
Card F gives:  
\( x = 6 \), \( y = 21 \).

Using \( y = kx \), solve for \( k \):

\[
k = \frac{y}{x} = \frac{21}{6} = 3.5
\]

Thus, the direct variation equation for card F is:

\[
y = 3.5x
\]

### 6. Which card can be represented by the equation \( y = \frac{1}{3}x \)?
We need to find which card has a constant of proportionality \( k = \frac{1}{3} \):

- Card B gives \( k = \frac{1}{3} \) (from earlier).

Thus, **Card B** is represented by the equation \( y = \frac{1}{3}x \).

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Would you like further explanation on any of the steps or details? Here are five related questions to expand on this:

1. What does the constant of proportionality represent in a direct variation?
2. How do you find the constant of proportionality when given multiple points?
3. Can a direct variation have a negative constant of proportionality?
4. How can you graph a direct variation equation?
5. What happens to the graph if the constant of proportionality is a fraction?

**Tip**: In direct variation problems, always look for the ratio \( \frac{y}{x} \) to identify the constant of proportionality, which stays the same for all points.

Solve the math problem involving direct variation based on the cards provided in the image. The questions involve finding the constant of proportionality, writing direct variation equations, and matching equations to the cards.

Solve direct variation problems by finding the constant of proportionality, writing equations, and matching direct variation equations to the provided data. This exercise is suitable for Grade 8 students and covers key algebraic concepts.

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