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The given problem states that the perimeter \( P \) of a rectangle with length \( \ell \) and width \( w \) is represented by the equation:

\[
P = 2\ell + 2w
\]

We need to find the expression that represents the width \( w \) in relation to \( P \) and \( \ell \).

### Step-by-step Solution:

1. Start with the equation for the perimeter:

\[
P = 2\ell + 2w
\]

2. Isolate \( w \) by subtracting \( 2\ell \) from both sides:

\[
P - 2\ell = 2w
\]

3. Now, solve for \( w \) by dividing both sides by 2:

\[
w = \frac{P - 2\ell}{2}
\]

This matches option **J**.

Thus, the correct answer is **J**: \(\frac{P - 2\ell}{2}\).

Would you like further details or have any questions? Here are some additional related questions:

1. How do we find the length if the width and perimeter are known?
2. How does the equation for the perimeter change if the rectangle becomes a square?
3. What would the expression for width be if only the area and length were given instead of the perimeter?
4. Can you explain why we divide by 2 to solve for the width?
5. How would the formula change if the units for length and width were different?

**Tip:** Whenever solving for a variable, always try to isolate it step by step using basic algebraic operations like addition, subtraction, multiplication, or division.

The perimeter P of a rectangle with length ℓ and width w can be represented by the equation P = 2ℓ + 2w. Which expression represents the width in relation to the P and ℓ?

Learn how to express the width of a rectangle in terms of its perimeter and length using the formula P = 2ℓ + 2w. This problem involves solving linear equations, algebraic manipulation, and understanding rectangle properties. Suitable for students in Grades 6-8.

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