Simplify The Expression 5-(2x-y)/(2x+y) With Detailed Steps

This article delves into the intricacies of the algebraic expression $5-\frac{2x-y}{2x+y}$, providing a step-by-step explanation of how to simplify and analyze it. Whether you're a student grappling with algebraic manipulations or simply seeking to enhance your mathematical understanding, this guide offers a comprehensive breakdown of the concepts involved. We will explore the fundamental principles of fraction manipulation, common denominators, and simplification techniques, ultimately empowering you to confidently tackle similar algebraic challenges.

Understanding the Expression

To effectively tackle the expression $5-\frac{2x-y}{2x+y}$, it's crucial to understand its structure and the operations involved. The expression consists of two main parts: the whole number 5 and the fraction $\frac{2x-y}{2x+y}$. The minus sign between them indicates subtraction. Therefore, the core task is to subtract the fraction from the whole number. This requires finding a common denominator and combining the terms appropriately.

Before diving into the simplification process, let's break down the components of the fraction itself. The numerator, $2x-y$, represents a linear expression involving two variables, x and y. Similarly, the denominator, $2x+y$, is another linear expression with the same variables. The fraction signifies the ratio between these two expressions. Understanding the nature of these components is essential for navigating the simplification process effectively. For example, we can analyze the behavior of the expression as x and y vary, identifying potential singularities or points where the denominator becomes zero, making the expression undefined. This deeper understanding not only aids in simplification but also provides valuable insights into the mathematical properties of the expression.

Furthermore, recognizing the absence of common factors between the numerator and denominator initially suggests that direct simplification by cancellation might not be immediately possible. This observation directs our focus towards manipulating the expression through common denominators and combining terms to potentially reveal hidden simplifications or to express the result in a more manageable form. The initial assessment of the expression's structure sets the stage for a strategic approach to its simplification, ensuring that each step is purposeful and contributes to the overall goal of obtaining the most reduced and understandable form.

Finding a Common Denominator

In order to subtract the fraction $\frac{2x-y}{2x+y}$ from the whole number 5, we need to express 5 as a fraction with the same denominator as the given fraction. This is a fundamental principle of fraction arithmetic. The denominator of the fraction is $2x + y$, so we need to rewrite 5 with this denominator. To achieve this, we multiply 5 by $\frac{2x+y}{2x+y}$, which is equivalent to multiplying by 1 and doesn't change the value of 5.

This process yields the expression $5 * \frac{2x+y}{2x+y} = \frac{5(2x+y)}{2x+y}$. Now, both terms in our original expression have the same denominator, $2x+y$. This allows us to combine them into a single fraction. The next step involves distributing the 5 in the numerator to expand the expression. By doing so, we transform the expression into a form where we can directly subtract the numerators, which is a crucial step in simplifying the overall expression. This strategic manipulation is key to making the subtraction operation feasible and moving towards a more concise representation of the original equation.

By carefully choosing the common denominator and rewriting the whole number 5, we've paved the way for a straightforward subtraction of the fractions. This approach highlights the importance of understanding the underlying principles of fraction arithmetic and applying them strategically to simplify complex algebraic expressions. This initial step of finding a common denominator is not just a mechanical process; it's a critical decision that sets the stage for the subsequent simplification steps and ultimately determines the ease with which we can arrive at the final answer.

Combining the Fractions

Now that we have a common denominator, we can combine the terms. We have the expression $\frac{5(2x+y)}{2x+y} - \frac{2x-y}{2x+y}$. To combine these fractions, we subtract the numerators while keeping the common denominator. This gives us $\frac{5(2x+y) - (2x-y)}{2x+y}$. This step is crucial as it consolidates the two terms into a single fraction, which is a significant simplification.

Next, we need to simplify the numerator by distributing the 5 and handling the subtraction carefully. It's important to pay close attention to the signs when subtracting the second term, $2x-y$. The negative sign in front of the parenthesis means we need to distribute it to both terms inside the parenthesis. This distribution is a common area for errors, so careful attention to detail is essential. Expanding the numerator, we get $10x + 5y - 2x + y$. Notice how the $-(-y)$ becomes $+y$. This step demonstrates the importance of understanding the rules of algebraic manipulation and applying them correctly to avoid mistakes.

By carefully combining the fractions and expanding the numerator, we've taken another significant step towards simplifying the expression. This process highlights the importance of methodical application of algebraic principles and careful attention to detail, particularly when dealing with signs and distributions. The resulting single fraction is now in a form that is easier to further simplify and analyze, setting the stage for the final steps of the simplification process.

Simplifying the Numerator

After combining the fractions, our expression is $\frac{10x + 5y - 2x + y}{2x+y}$. The next step is to simplify the numerator by combining like terms. In the numerator, we have terms with 'x' and terms with 'y'. Combining the 'x' terms, we have $10x - 2x = 8x$. Combining the 'y' terms, we have $5y + y = 6y$.

This simplification process is a fundamental aspect of algebraic manipulation. By identifying and combining like terms, we reduce the complexity of the expression and make it easier to understand and work with. This step not only simplifies the numerator but also potentially reveals opportunities for further simplification, such as factoring out common factors. The ability to identify and combine like terms efficiently is a crucial skill in algebra and is essential for solving equations and simplifying expressions.

Now, our expression becomes $\frac{8x + 6y}{2x+y}$. This simplified numerator provides a clearer picture of the relationship between the variables x and y in the expression. The next logical step is to examine the simplified numerator and denominator to see if there are any common factors that can be factored out and potentially canceled, leading to further simplification of the expression. This methodical approach, where we simplify each part of the expression step-by-step, is key to successfully tackling complex algebraic problems.

Factoring and Final Simplification

Our expression is now $\frac{8x + 6y}{2x+y}$. To further simplify, we look for common factors in the numerator and the denominator. In the numerator, both $8x$ and $6y$ are divisible by 2. Factoring out a 2 from the numerator, we get $2(4x + 3y)$. The denominator, $2x+y$, does not have any immediately obvious factors.

This factoring step is a crucial technique in simplifying algebraic expressions. By identifying and factoring out common factors, we can often reduce the expression to its simplest form. In this case, factoring out the 2 from the numerator reveals a potential for cancellation if the denominator shared a similar factor. However, upon inspection, we see that the expression inside the parenthesis in the numerator, $4x + 3y$, is different from the denominator, $2x + y$. This indicates that we cannot directly cancel any terms.

Therefore, the simplified expression is $\frac{2(4x + 3y)}{2x+y}$. Since there are no further common factors between the numerator and the denominator, this is the final simplified form of the expression. This result highlights the importance of thorough factorization and careful comparison of terms to ensure that the expression is indeed in its simplest form. While we were unable to achieve a further reduction through cancellation, the factoring step was essential in confirming that the expression is now in its most concise representation. This final simplified form is not only easier to work with but also provides a clearer understanding of the relationships between the variables x and y in the original expression.

Final Answer

After performing all the necessary steps, we have simplified the expression $5-\frac{2x-y}{2x+y}$ to $\frac{2(4x + 3y)}{2x+y}$. This is the final simplified form of the expression. We achieved this by finding a common denominator, combining the fractions, simplifying the numerator, and factoring out common factors. This process demonstrates the key principles of algebraic manipulation and provides a clear and concise solution to the problem.

This final answer represents the culmination of a series of strategic simplification steps. Each step, from finding the common denominator to factoring out common factors, played a crucial role in reducing the complexity of the original expression. The resulting simplified form not only provides a more manageable representation of the expression but also offers valuable insights into the relationship between the variables x and y. The process of simplifying algebraic expressions is not just about finding the answer; it's about developing a deeper understanding of mathematical principles and techniques that can be applied to a wide range of problems. By mastering these techniques, one can confidently tackle more complex algebraic challenges and gain a greater appreciation for the elegance and power of mathematics.