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.