Problem 86 - Intermediate Algebra

Question

Answer the question with an algebraic expression. The perimeter of a square is \(c\) centimeters. How long is each side of the square?

Step-by-Step Solution

Answer

Each side of the square is \( \frac{c}{4} \) centimeters long.

Step 1: Identify Perimeter Formula

The perimeter of a square is calculated by adding all four sides. Since all sides of a square are equal, the formula for the perimeter is given by \( P = 4s \), where \( s \) is the length of one side.

Step 2: Set Up Equation

We know from the problem statement that the perimeter of the square is \( c \) centimeters. So, we can set up the equation for the perimeter as \( 4s = c \).

Step 3: Solve for Side Length

To find the length of each side, we need to solve the equation \( 4s = c \) for \( s \). We do this by dividing both sides of the equation by 4, resulting in \( s = \frac{c}{4} \).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are a crucial part of mathematics. They help us represent real-world problems in a mathematical format. An algebraic expression is a combination of numbers, variables, and operation symbols. For example, the perimeter of a square can be expressed as the algebraic expression \(P = 4s\). Here, \(P\) is the perimeter, and \(s\) is the length of one side of the square.

This expression shows us that the perimeter is four times the length of one side. Algebraic expressions allow us to create equations by substituting known values, which we can then solve for unknown quantities.

Solving for Side Length

When dealing with squares, each side is the same length. Knowing the perimeter, or total length around the square, helps us find the length of one side. Given \(P = 4s\), if we know \(P = c\), we substitute \(c\) into the equation, where \(c\) is the perimeter.

This gives us the equation \(4s = c\). Solving for \(s\) requires isolating \(s\) on one side of the equation. This rearrangement will tell us the length of one side in terms of the perimeter.

Dividing Equations to Solve for Variables

To solve the equation \(4s = c\) for \(s\), division is a key tool. Dividing both sides of the equation by 4 will isolate \(s\). This step is crucial because it transforms the equation, making \(s\) the subject.

After dividing, the new equation is \(s = \frac{c}{4}\). Here's why dividing works:

• Division is the mathematical operation that "undoes" multiplication. Since \(4s\) means four times \(s\), dividing by 4 reverses this.
• Applying division equally to both sides maintains the equation's balance, which is a fundamental principle of algebra.

This approach results in an expression for the side length in terms of the perimeter, making problems like these much simpler to manage.