Calculating The Difference Between Length And Width Of A Rectangular Field

In the realm of mathematics, particularly in algebra and geometry, we often encounter problems that require us to manipulate expressions and solve for unknowns. One such problem involves finding the difference between the length and width of a rectangular field, given their respective expressions. This article delves into a step-by-step solution to this problem, providing a comprehensive understanding of the underlying concepts and techniques. We will explore how to subtract algebraic expressions, simplify the result, and arrive at the correct answer. By the end of this article, you will not only be able to solve this specific problem but also gain a broader understanding of how to approach similar mathematical challenges. The key to success in mathematics lies in understanding the fundamental principles and applying them methodically. This article aims to provide that understanding and equip you with the skills to tackle various algebraic problems.

The length of a rectangular field is represented by the expression $14x - 3x^2 + 2y$, and the width of the field is represented by the expression $5x - 7x^2 + 7y$. The question asks: How much greater is the length of the field than the width?

This problem falls under the category of algebra, specifically dealing with the manipulation of polynomial expressions. To solve this, we need to subtract the expression representing the width from the expression representing the length. This involves combining like terms and simplifying the resulting expression. Understanding the concept of like terms – terms that have the same variables raised to the same powers – is crucial for this task. For instance, $14x$ and $5x$ are like terms, as are $-3x^2$ and $-7x^2$, and $2y$ and $7y$. The process of combining like terms involves adding or subtracting their coefficients while keeping the variable part the same. Mastering this skill is essential for simplifying algebraic expressions and solving equations. This problem provides an excellent opportunity to practice and reinforce this fundamental concept.

To determine how much greater the length is than the width, we need to subtract the expression for the width from the expression for the length. This can be written as:

$(14x - 3x^2 + 2y) - (5x - 7x^2 + 7y)$

The first step in solving this is to distribute the negative sign to each term inside the second parenthesis. This changes the signs of the terms within the parenthesis, effectively turning addition into subtraction and vice versa. This is a crucial step as it ensures that we are subtracting the entire expression for the width, not just the first term. The expression now becomes:

$14x - 3x^2 + 2y - 5x + 7x^2 - 7y$

Next, we need to identify and combine like terms. As mentioned earlier, like terms are those that have the same variables raised to the same powers. In this expression, we have three sets of like terms: terms with $x$, terms with $x^2$, and terms with $y$. Let's group them together:

$(14x - 5x) + (-3x^2 + 7x^2) + (2y - 7y)$

Now, we can combine the coefficients of the like terms:

• For the terms with $x$: $14x - 5x = 9x$
• For the terms with $x^2$: $-3x^2 + 7x^2 = 4x^2$
• For the terms with $y$: $2y - 7y = -5y$

Combining these results, we get the simplified expression:

$9x + 4x^2 - 5y$

Therefore, the length of the field is $9x + 4x^2 - 5y$ greater than the width. This result represents the difference between the length and width in terms of the variables $x$ and $y$. The final expression is a simplified polynomial, which provides a clear and concise representation of the difference. Understanding how to perform these algebraic manipulations is fundamental to solving a wide range of mathematical problems.

Comparing our result, $9x + 4x^2 - 5y$, with the given options, we find that it matches option A.

Therefore, the answer is:

A. $9x + 4x^2 - 5y$

This answer confirms that our step-by-step solution was accurate. It highlights the importance of careful attention to detail when performing algebraic manipulations. A common mistake is to incorrectly distribute the negative sign or to misidentify like terms. By following a systematic approach, as demonstrated in this solution, we can minimize the risk of errors and arrive at the correct answer. This problem serves as a valuable example of how algebraic principles can be applied to solve practical problems involving geometric shapes and their dimensions.

It's important to analyze the distractors (incorrect answer choices) to understand why they might be chosen and what common errors they represent. In this case, let's consider why option B, $9x - 10x^2 - 5y$, might be a distractor.

Option B, $9x - 10x^2 - 5y$, shares some similarities with the correct answer, which indicates that it likely arises from a mistake in the process of combining like terms. Specifically, the difference in the coefficient of the $x^2$ term suggests an error in adding or subtracting the coefficients of $-3x^2$ and $7x^2$. A possible error could be incorrectly subtracting 7 from -3 instead of adding, leading to $-3 - 7 = -10$ instead of $-3 + 7 = 4$. This highlights the importance of careful attention to the signs of the coefficients when combining like terms. Another potential error could be a misunderstanding of the rules of adding and subtracting integers. This distractor serves as a reminder to double-check each step in the process, particularly when dealing with negative signs. By understanding the common errors that lead to distractors, we can develop strategies to avoid them and improve our problem-solving accuracy. Analyzing distractors is a valuable learning tool that can enhance our understanding of the concepts and techniques involved.

This problem reinforces several key concepts in algebra:

1. Subtracting Algebraic Expressions: The core of the problem involves subtracting one algebraic expression from another. This requires distributing the negative sign and then combining like terms.
2. Combining Like Terms: Like terms are terms that have the same variables raised to the same powers. Combining them involves adding or subtracting their coefficients.
3. Distributive Property: The distributive property is crucial when subtracting expressions in parentheses. It ensures that the negative sign is applied to all terms within the parentheses.
4. Polynomial Expressions: The problem deals with polynomial expressions, which are expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication.

The key takeaways from this problem are:

• Pay attention to signs: Carefully track the signs of the coefficients when adding, subtracting, and distributing.
• Combine like terms correctly: Ensure that you are only combining terms with the same variables raised to the same powers.
• Distribute negative signs properly: When subtracting an expression in parentheses, distribute the negative sign to each term inside.
• Check your work: After simplifying an expression, review your steps to ensure accuracy.

By mastering these concepts and applying these takeaways, you can confidently solve a wide range of algebraic problems. This problem serves as a foundation for more advanced topics in algebra and calculus.

While this problem is presented in a mathematical context, the underlying concepts have real-world applications. For instance, consider a scenario where you are planning a garden. The length and width of the garden can be represented by algebraic expressions, and you might need to calculate the difference between the dimensions to determine the amount of fencing required or the area available for planting. Algebraic expressions are also used in various fields such as engineering, physics, economics, and computer science to model and solve problems. For example, engineers might use algebraic expressions to represent the dimensions of a structure, physicists might use them to describe the motion of an object, and economists might use them to model market trends. Understanding how to manipulate and simplify algebraic expressions is therefore a valuable skill that can be applied in a variety of contexts. This problem, while seemingly abstract, provides a foundation for applying mathematical principles to real-world situations.

In conclusion, finding the difference between the length and width of a rectangular field, when given their algebraic expressions, is a problem that highlights the importance of fundamental algebraic concepts. By carefully subtracting the expressions, combining like terms, and simplifying the result, we arrived at the correct answer. This problem not only tests our ability to perform algebraic manipulations but also reinforces the significance of paying attention to detail and avoiding common errors. The skills and concepts learned in solving this problem are applicable to a wide range of mathematical and real-world scenarios. Mastering these skills will undoubtedly enhance your problem-solving abilities and open doors to further exploration in mathematics and related fields.