The formula for the perimeter $P$ of a rectangle is given by:

$P = 2 \times (\text{length} + \text{width})$

Here, the width $W$ of the rectangle is $(5v - 2w)$ cm, and the length $L$ is $(6v + 7w)$ cm. Plugging these expressions into the perimeter formula:

$P = 2 \times \left[(6v + 7w) + (5v - 2w)\right]$

Step-by-step Solution

1. Add the length and the width:

$(6v + 7w) + (5v - 2w) = 6v + 5v + 7w - 2w$

2. Combine like terms:

$6v + 5v = 11v \quad \text{and} \quad 7w - 2w = 5w$

So, the sum of the length and width is:

$11v + 5w$

3. Multiply by 2 to find the perimeter:

$P = 2 \times (11v + 5w) = 22v + 10w$

Thus, the expression that represents the perimeter of the rectangle is:

$22v + 10w \text{ centimeters}$

Would you like a detailed explanation of any part of this solution, or do you have any questions?

Here are 5 related questions to expand your understanding:

1. How would the expression for the perimeter change if the width was $(7v - 3w)$ and the length was $(4v + 5w)$?
2. How would you express the area of the rectangle in terms of $v$ and $w$?
3. What is the perimeter if $v = 3$ cm and $w = 2$ cm?
4. If the perimeter of the rectangle is given, how can you find the values of $v$ and $w$?
5. How does changing the values of $v$ and $w$ affect the perimeter?

Tip: Always combine like terms before performing arithmetic operations when simplifying algebraic expressions.