Let's analyze the problem carefully to determine which of the given representations are correct.

Carson starts by running 2 miles each day, then walks at a rate of 4 miles per hour. We want to find the total number of miles $y$ he covers in a day, based on the number of hours $x$ he spends walking.

The equation for this situation can be written as: $y = 2 + 4x$ where:

• $2$ represents the miles he runs at the beginning of the day, and
• $4x$ represents the miles he covers walking at a rate of 4 miles per hour for $x$ hours.

Now, let's evaluate the options:

1. Option A: $y = 2x + 4$
   This equation is incorrect because it suggests that the rate of walking is 2 miles per hour, and the constant distance covered is 4 miles. This does not match the situation described.

2. Option B: Number of Hours Walked - Total Number of Miles Table (1, 6; 2, 10)
   If $x = 1$, then $y = 2 + 4(1) = 6$.
   If $x = 2$, then $y = 2 + 4(2) = 10$.
   This matches the table, so this is a correct representation.

3. Option C: (No clear content, but if it matches $y = 2 + 4x$, it could be correct.)
   We need more information to assess Option C.

4. Option D: $y = 4x + 2$
   This equation correctly represents the situation: 4 miles per hour walking rate multiplied by the number of hours $x$, plus 2 miles that he starts with. This is correct.

5. Option E: Number of Hours Walked - Total Number of Miles Table (1, 6; 2, 8)
   If $x = 1$, then $y = 6$, which is correct. But if $x = 2$, then $y = 10$, not 8. Therefore, this is incorrect.

6. Option F: (No content provided here, so cannot be evaluated.)

Correct Answers

Based on the analysis:

• B (Correct table)
• D (Correct equation)
• If more information aligns with $y = 4x + 2$, possibly C or F could be correct.

Would you like further details or have any questions?

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Related Questions:

1. What would the equation be if Carson runs 3 miles instead of 2?
2. How would the graph of $y = 4x + 2$ look like on a coordinate plane?
3. How can we determine if a table correctly represents a linear equation?
4. What happens to the total distance $y$ if Carson increases his walking speed?
5. How would the equation change if Carson also bikes at a different rate?

Tip:

Always substitute values back into an equation to verify if a given representation (table or equation) matches the described situation.