Let's analyze the given equation $P = 2(l + w)$ and see which of the provided options is equivalent.

Step-by-step simplification of $P = 2(l + w)$:

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

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

3. To solve for $w$, subtract $l$ from both sides: $w = \frac{P}{2} - l$

Now, let's compare this result with the given options:

• Option 1: $w = P - 2l$
  This is incorrect because it does not account for the division by 2 in $P$.

• Option 2: $2 = P - l$
  This is incorrect because it does not isolate $w$ and does not resemble the original equation.

• Option 3: $w = \frac{P - 2l}{2}$
  This is correct because it matches the equation we derived after simplifying $w = \frac{P}{2} - l$.

• Option 4: $w = \frac{P + 2l}{2}$
  This is incorrect because it adds $2l$ instead of subtracting it.

Conclusion:

The correct equivalent equation is: $w = \frac{P - 2l}{2}$

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How would the equation change if you wanted to solve for $l$ instead of $w$?
2. What does the variable $P$ represent in this equation?
3. How can you interpret this equation geometrically for a rectangle?
4. What happens to the value of $w$ if both $P$ and $l$ are doubled?
5. How would you solve this equation if you had specific values for $P$ and $l$?

Tip: Always check your operations step-by-step when manipulating equations to avoid mistakes with signs and fractions!