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The perimeter of a rectangle is the total distance around the outside of the rectangle. It can be calculated using the lengths of the sides. For rectangles, the formula to find the perimeter is:

• \( P = 2(l + w) \)

where \( l \) is the length and \( w \) is the width. In this exercise, the length \( l \) is given by \( 3x + 3 \) and the width \( w \) is \( \frac{1}{3} \). Adding these together forms the expression inside the parentheses:

• \( 3x + 3 + \frac{1}{3} = 3x + \frac{9}{3} + \frac{1}{3} = 3x + \frac{10}{3} \)

Then, using the perimeter formula, substitute this expression:

• \( P = 2(3x + \frac{10}{3}) \)

Simplifying gives:

• \( P = 6x + \frac{20}{3} \)

This expression shows how the perimeter changes as the variable \( x \) changes.

The area of a rectangle is the amount of space enclosed within its sides and is calculated by multiplying the length by the width. The formula used is:

• \( A = l \times w \)

In this scenario, substituting the given expressions, \( l = 3x + 3 \) and \( w = \frac{1}{3} \), into the formula gives:

• \( A = (3x + 3) \times \frac{1}{3} \)

By distributing the \( \frac{1}{3} \) across the terms in the expression:

• \( A = \frac{1}{3} \times 3x + \frac{1}{3} \times 3 \)
• \( A = x + 1 \)

This simplified expression \( A = x + 1 \) shows how the area will vary depending on the values of \( x \).

Variable expressions allow us to write relationships using letters, often called variables, to represent numbers. This makes them useful in algebra. In this exercise, the variable \( x \) is used to express both the length and width of a rectangle. The expression for length is \( l = 3x + 3 \), showing that it is directly affected by \( x \). Similarly, the width has a constant term as \( w = \frac{1}{3} \).Variable expressions convert a fixed formula into something flexible that can adapt to different scenarios by changing the value of \( x \). This lets you calculate outcomes like the perimeter and area for any number that \( x \) might take. By substituting \( x \) into these expressions, it becomes clear how both the perimeter and area change in response to the value of \( x \) shifting.This approach helps in visualizing how algebra uses mathematical operations to express real-world scenarios, offering insights into relationships between different quantities such as dimensions of shapes.