The equation given for the perimeter of a rectangle is:

$P = 2(l + w)$

We are asked to find the equation equivalent to this that isolates $w$.

Step-by-step process:

1. Start with the perimeter equation: $P = 2(l + w)$

2. Divide both sides of the equation by 2 to get rid of the factor of 2: $\frac{P}{2} = l + w$

3. Now subtract $l$ from both sides to isolate $w$: $w = \frac{P}{2} - l$

This is the equation for $w$.

Checking the options:

The equivalent equation is:

$w = \frac{P}{2} - l$

This matches the third option in the list provided in the image.

Let me know if you'd like any further explanation or if you have questions! Here are 5 related questions to expand the understanding:

1. How do you derive the formula for the perimeter of a rectangle?
2. How would you solve for $l$ in the perimeter formula instead of $w$?
3. What are some other real-world applications of the perimeter formula?
4. What is the difference between the perimeter and the area of a rectangle?
5. How would the equation change if the perimeter formula was for a square instead of a rectangle?

Tip: Always check each step for accuracy when manipulating equations, especially when dividing or multiplying both sides!