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To solve this problem, let’s break down the given information:

1. **Dimensions of the Pool**: The pool is rectangular with dimensions 6 meters by 4 meters.
   - The **area of the pool** is \(6 \times 4 = 24\) square meters.

2. **Walkway Around the Pool**: The walkway surrounds the pool with a width \( x \) meters.
   - This means the overall dimensions of the pool plus walkway are:
     - **Length**: \(6 + 2x\) (since \(x\) is added to both sides)
     - **Width**: \(4 + 2x\) (since \(x\) is added to both sides)

3. **Area of Pool + Walkway**:
   - The area of the rectangle formed by the pool and walkway combined is:
     \[
     (6 + 2x)(4 + 2x) = 24 + 12x + 8x + 4x^2 = 24 + 20x + 4x^2
     \]

4. **Area of the Walkway Alone**:
   - To find the area of the walkway alone, we subtract the area of the pool from the combined area:
     \[
     \text{Area of walkway} = (24 + 20x + 4x^2) - 24 = 20x + 4x^2
     \]

5. **Setting the Walkway Area Equal to Pool Area**:
   - We are asked to find the value of \(x\) for which the area of the walkway equals the area of the pool:
     \[
     20x + 4x^2 = 24
     \]

6. **Solving the Equation**:
   - Rearranging the equation, we get:
     \[
     4x^2 + 20x - 24 = 0
     \]
   - Divide the entire equation by 4 to simplify:
     \[
     x^2 + 5x - 6 = 0
     \]
   - Now, we factor this quadratic equation:
     \[
     (x + 6)(x - 1) = 0
     \]
   - Setting each factor to zero gives:
     \[
     x = -6 \quad \text{or} \quad x = 1
     \]
   - Since \(x\) represents a width, it must be positive, so we discard \(x = -6\).

Therefore, the value of \(x\) that makes the area of the walkway equal to the area of the pool is:
\[
x = 1 \text{ meter}
\]

Would you like a detailed explanation of any step? 

Here are some related questions you might consider:

1. What if the pool was circular instead of rectangular? How would the calculation change?
2. How would the solution change if the walkway width varied on each side of the pool?
3. What if the pool had different dimensions, say 8 meters by 5 meters? How would that affect the value of \(x\)?
4. What formula is used to find the area around a shape with a uniform width walkway?
5. Can this problem be solved using a graphical method?

**Tip**: When solving problems involving areas, always clearly distinguish the area of the object itself and the additional areas around it.

A rectangular pool 6 meters by 4 meters is surrounded by a walkway of width x meters. At what value of x will the area of the walkway equal the area of the pool?

Learn how to find the width of a walkway surrounding a rectangular pool where the walkway's area equals the area of the pool. This math problem involves algebra and quadratic equations, suitable for Grades 9-10.

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