Solving Rectangle Width Problem Equation And Step-by-Step Guide

In this comprehensive guide, we will delve into the process of determining the width of a rectangle when its perimeter and the relationship between its length and width are known. We will break down the problem step by step, ensuring a clear understanding of the underlying concepts and the algebraic techniques involved. By the end of this article, you will be equipped with the knowledge and skills to solve similar geometric problems with confidence.

Understanding the Problem

Before we jump into the equation, let's first understand the problem statement. We have a rectangle where the length is 6 inches longer than the width. The perimeter of this rectangle is given as 28 inches. Our goal is to find an equation that can help us determine the value of 'x', which represents the width of the rectangle in inches. This is a classic problem in geometry and algebra, combining the properties of rectangles with algebraic equation solving. The key to solving this problem lies in translating the word problem into a mathematical equation that accurately represents the given information.

Setting up the Variables

In this mathematical puzzle, let's begin by defining our variables. We are given that 'x' represents the width of the rectangle. Since the length is 6 inches longer than the width, we can express the length as 'x + 6'. These expressions are crucial as they form the foundation of our equation. By clearly defining these variables, we can translate the word problem into a mathematical representation. This is a fundamental step in solving any algebraic problem, as it allows us to work with symbols and equations rather than descriptive words.

Recalling the Perimeter Formula

To proceed, we need to recall the formula for the perimeter of a rectangle. The perimeter is the total distance around the rectangle, which is calculated by adding up the lengths of all four sides. For a rectangle, this is given by the formula: Perimeter = 2 * (length + width). This formula is the cornerstone of our solution, as it provides the link between the dimensions of the rectangle and its perimeter. Understanding and correctly applying this formula is essential for solving the problem.

Constructing the Equation

Translating Words into Math

Now, let's translate the given information into a mathematical equation. We know the perimeter is 28 inches, the width is 'x', and the length is 'x + 6'. Using the perimeter formula, we can write: 28 = 2 * ((x + 6) + x). This equation represents the relationship between the perimeter, length, and width of the rectangle. It is a crucial step in solving the problem, as it allows us to use algebraic techniques to find the value of 'x'.

Simplifying the Equation

The equation we've constructed, 28 = 2 * ((x + 6) + x), needs to be simplified to make it easier to solve. First, we simplify the expression inside the parentheses: (x + 6) + x becomes 2x + 6. Then, we multiply the entire expression by 2: 2 * (2x + 6) becomes 4x + 12. So, our equation now looks like this: 28 = 4x + 12. This simplified form is much easier to work with and allows us to isolate the variable 'x'.

Solving for x

Isolating the Variable

To find the value of 'x', we need to isolate it on one side of the equation. We start with our simplified equation: 28 = 4x + 12. The first step is to subtract 12 from both sides of the equation: 28 - 12 = 4x + 12 - 12, which simplifies to 16 = 4x. Now, we have the term with 'x' isolated on one side. This process of isolating the variable is a fundamental technique in algebra, allowing us to determine its value.

Determining the Value of x

Now that we have 16 = 4x, we can solve for 'x' by dividing both sides of the equation by 4: 16 / 4 = 4x / 4. This simplifies to 4 = x. Therefore, the width of the rectangle, represented by 'x', is 4 inches. This is the solution to our problem, and it represents the numerical value of the width based on the given information.

Verifying the Solution

Checking the Answer

It's always a good practice to verify our solution to ensure it's correct. We found that x = 4 inches, which means the width is 4 inches. The length, being 6 inches longer, is 4 + 6 = 10 inches. Now, we can use the perimeter formula to check: Perimeter = 2 * (length + width) = 2 * (10 + 4) = 2 * 14 = 28 inches. This matches the given perimeter, so our solution is correct. This verification step is crucial as it confirms the accuracy of our calculations and ensures that the solution aligns with the problem's conditions.

Ensuring Accuracy

By verifying our solution, we not only confirm its correctness but also reinforce our understanding of the problem and the concepts involved. This process helps to build confidence in our problem-solving skills and ensures that we are applying the correct methods. In mathematics, accuracy is paramount, and verification is a key tool in achieving it.

Identifying the Correct Equation

Matching the Steps

Now that we've gone through the process of solving for the width, let's identify which of the given equations can be used to find the value of 'x'. We derived the equation 28 = 2 * ((x + 6) + x), which simplifies to 28 = 4x + 12. Looking at the given options, we need to find the one that is equivalent to this equation. This step is crucial as it connects our derived solution with the provided options, ensuring we select the correct equation.

Selecting the Right Option

Among the given options, the equation that matches our derived equation or its equivalent form is the correct one. The process of simplifying and solving the equation helps us to identify the correct option with confidence. This step highlights the importance of algebraic manipulation in problem-solving, as it allows us to transform equations into different forms while preserving their equality.

Conclusion

Recap of the Solution

In this detailed guide, we successfully determined the equation needed to find the width of a rectangle given its perimeter and the relationship between its length and width. We started by understanding the problem, setting up variables, and recalling the perimeter formula. Then, we constructed and simplified the equation, solved for 'x', and verified our solution. Finally, we identified the correct equation from the given options. This step-by-step approach ensures a clear and comprehensive understanding of the problem-solving process.

Importance of Step-by-Step Problem Solving

This exercise demonstrates the importance of a methodical, step-by-step approach to solving mathematical problems. By breaking down the problem into smaller, manageable steps, we can tackle even complex problems with confidence. This approach not only helps in finding the correct solution but also enhances our understanding of the underlying concepts. In mathematics, a systematic approach is often the key to success.

Final Thoughts

By mastering the techniques discussed in this guide, you will be well-equipped to solve a wide range of geometric and algebraic problems. Remember, the key is to understand the problem, translate it into mathematical terms, and apply the appropriate formulas and techniques. With practice and a methodical approach, you can excel in problem-solving and develop a deeper appreciation for the power of mathematics. This skill enhancement is invaluable in both academic and real-world applications.