Exercises 141–143 will help you prepare for the material covered in the next section. If the width of a rectangle is represented by x and the length is represented by x + 200, write a simplified algebraic expression that models the rectangle's perimeter.

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Start by recalling the formula for the perimeter of a rectangle: P = 2(length) + 2(width).

Substitute the given expressions for the length and width into the formula. The width is x, and the length is x + 200. This gives P = 2(x + 200) + 2(x).

Distribute the 2 across the terms inside the parentheses for the first term: 2(x + 200) becomes 2x + 400.

Combine the distributed term with the second term: P = 2x + 400 + 2x.

Simplify the expression by combining like terms: P = (2x + 2x) + 400, which simplifies to P = 4x + 400.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Perimeter of a Rectangle

The perimeter of a rectangle is the total distance around the rectangle, calculated by adding together the lengths of all four sides. The formula for the perimeter (P) is P = 2(length + width). Understanding this formula is essential for modeling the perimeter based on given dimensions.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. In this context, the expression will involve the variable x, representing the width, and will be used to express the perimeter in terms of x. Simplifying these expressions is a key skill in algebra.

Variable Representation

In algebra, variables are symbols used to represent unknown values or quantities. In this problem, x represents the width of the rectangle, while x + 200 represents the length. Understanding how to manipulate and substitute variables is crucial for forming and simplifying algebraic expressions.