https://file.aiquickdraw.com/mathbot/file/17261914360990oo8qknh.jpg

To solve the problem, let's analyze the given equation and the choices provided:

### Problem Breakdown:
We are given that:
\[ \frac{x}{y} = \frac{3}{7} \]

This implies:
\[ x = \frac{3}{7}y \]

The question asks which of the following values **cannot** be a possible value of \( y - x \).

### Steps:

1. **Express \( y - x \) in terms of \( y \):**
   \[
   y - x = y - \frac{3}{7}y
   \]
   \[
   y - x = \frac{7y - 3y}{7} = \frac{4y}{7}
   \]

   Therefore:
   \[
   y - x = \frac{4}{7}y
   \]

2. **Check each option to see if it can be written as \( \frac{4}{7}y \):**

   - (A) **4**: 
     \[
     \frac{4}{7}y = 4 \Rightarrow y = 4 \times \frac{7}{4} = 7
     \]
     \( y = 7 \) is a possible value.
   
   - (B) **21**:
     \[
     \frac{4}{7}y = 21 \Rightarrow y = 21 \times \frac{7}{4} = 36.75
     \]
     \( y = 36.75 \) is also a possible value.
   
   - (C) **24**:
     \[
     \frac{4}{7}y = 24 \Rightarrow y = 24 \times \frac{7}{4} = 42
     \]
     \( y = 42 \) is a possible value.

   - (D) **84**:
     \[
     \frac{4}{7}y = 84 \Rightarrow y = 84 \times \frac{7}{4} = 147
     \]
     \( y = 147 \) is a possible value.

   - (E) **None of these**: If none of the given options fit, this would be the answer. However, all other options have been verified to be possible.

### Conclusion:
All the given options \( 4, 21, 24, 84 \) can be possible values of \( y - x \), so the correct answer is:

**(E) None of these**

Would you like further details on the solution or have any other questions?

### Related Questions:

1. What happens if \( y = 1 \), how does \( y - x \) behave?
2. Can \( y - x \) ever be negative? Why or why not?
3. How would the problem change if \( \frac{x}{y} = \frac{5}{7} \)?
4. How can we use this relation to solve problems involving ratios?
5. What happens if \( y - x \) is a fraction, does it affect the solution?

### Tip:
When dealing with ratios, always express one variable in terms of the other to simplify and solve for the unknowns easily.

If x/y = 3/7, then which of the following cannot be a possible value of (y - x)? (A) 4, (B) 21, (C) 24, (D) 84, (E) None of these.

Learn how to solve a ratio problem where x/y = 3/7, and identify the possible values for y - x. Explore the algebraic concepts and formulas needed to find which values of y - x are valid. Suitable for students in Grades 8-10.

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