Problem 45 - Algebra 1

Question

Write an equation or an inequality to model the real-life situation. The perimeter \(P\) of a square is equal to four times the difference of a number \(s\) and two.

Step-by-Step Solution

Answer

The equation representing the perimeter of the square given the conditions in the problem statement is \(P = 4(s - 2)\). This equation can be solved for 's' to determine the length of the side of the square in a specific application of this model.

Step 1: Analyze the given situation

First, understand that the perimeter of a square is the sum of its four equal sides. Let the length of the side be represented by 's'.

Step 2: Express the Perimeter of the Square

The general mathematical formula for the perimeter of a square with side length 's' is \(P = 4s\). This is based on the property that all sides of a square are equal.

Step 3: Formulate the Equation from the Problem Statement

From the problem statement, we are told that the perimeter of the square is equal to four times the difference of a number 's' and two. We translate this into the mathematical expression \(P = 4(s - 2)\), with 's' representing the side length.

Key Concepts

Equations

Understanding how to create and solve equations is a cornerstone of algebra. An equation is a statement that two expressions are equal, and it often contains one or more variables. In the given exercise, the equation represents a real-life situation where the perimeter (\(P\)) of a square is equal to four times the difference of a number (\(s\) - a variable representing the side length) and two.

In setting up the equation, we are mapping the words of the problem statement into a mathematical format. The result, \(P = 4(s - 2)\), allows us to calculate the perimeter of the square when the value of \(s\) is known. By solving this equation for \(s\), we can find the length of the sides of the square based on its perimeter. This also illustrates the utility of equations in solving problems: by manipulating the equation, we manipulate the values the variables can take to find an answer to the original problem.

Inequalities

While equations assert that two expressions are exactly equal, inequalities state that one expression is greater or less than another. They are expressed using symbols like \(>\), \(<\), \(\geq\), or \(\leq\). Inequalities are crucial in scenarios where a range of solutions is acceptable, or constraints are imposed, like in optimizing problems.

Even though the original exercise revolves around an equation, imagine if we were instead asked to find a range where the perimeter of the square is larger than a certain value. We would have formulated an inequality such as \(P > 4(s - 2)\). Solving inequalities involves similar steps to solving equations, but we must keep in mind the direction of the inequality whenever we multiply or divide by negative numbers, as this action will flip the inequality sign.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables (like \(s\)), and arithmetic operations (addition, subtraction, multiplication, and division). Unlike equations, expressions do not have an equal sign. They are the building blocks of algebra and can represent quantitative relationships or general mathematical rules.

In our exercise, \(4(s - 2)\) is the algebraic expression that models the perimeter. It describes the quantity without setting it equal to anything. Expressions can be evaluated once we substitute numerical values for variables, which in this case would give us the perimeter for a specific square. The ability to create and interpret algebraic expressions is pivotal for students as it strengthens their problem-solving skills and understanding of mathematical relationships.

Mathematical Modeling

Mathematical modeling is the process of using mathematical language and tools to represent real-world situations. It allows us to predict, analyze, and understand various phenomena by creating a simplified version of reality that can be manipulated mathematically.

The exercise presents a simple instance of mathematical modeling. It takes the physical concept of a square's perimeter and translates it into a mathematical equation using the relevant variables and constants. By modeling the problem, one can use the equation \(P = 4(s - 2)\) to calculate unknown information about the square, given some initial data. Mathematical modeling is a skill that extends far beyond pure mathematics, finding applications in fields such as engineering, economics, biology, and social sciences.