Solving -4/(y+2) = 1/(5y+10) + 1 For Y A Step-by-Step Guide

Jul 12, 2025 by ADMIN 60 views

Solving for y in the Equation -(4/(y+2)) = (1/(5y+10)) + 1

Introduction

In this comprehensive guide, we will delve into the step-by-step process of solving the equation -4/(y+2) = (1/(5y+10)) + 1 for the variable y. This equation involves fractions and algebraic manipulation, requiring a systematic approach to arrive at the solution. Understanding how to solve such equations is crucial for various applications in mathematics, physics, and engineering. This article aims to provide a clear and detailed explanation, making it accessible even if you're not a math expert. By the end of this guide, you'll be well-equipped to tackle similar algebraic problems with confidence.

Understanding the Equation

Before diving into the solution, let's break down the equation -4/(y+2) = (1/(5y+10)) + 1. The equation involves rational expressions, which are fractions where the numerator and denominator are polynomials. In this case, we have fractions with y in the denominator. The goal is to isolate y on one side of the equation to find its value. The first step usually involves eliminating the fractions to simplify the equation. This can be done by finding a common denominator and multiplying both sides of the equation by it. We also need to be mindful of values of y that would make the denominator zero, as these values would make the expressions undefined. Let's embark on the journey of solving this equation step by step, ensuring clarity and accuracy in each stage.

Step-by-Step Solution

To solve the equation -4/(y+2) = (1/(5y+10)) + 1, we'll follow a methodical approach, ensuring each step is clear and logical. First, we need to simplify the equation by eliminating the fractions. This involves finding the least common denominator (LCD) of the denominators (y+2) and (5y+10). Notice that 5y+10 can be factored as 5(y+2). Therefore, the LCD is 5(y+2). Multiplying both sides of the equation by the LCD will clear the fractions. Be careful to distribute the multiplication correctly on both sides of the equation. This process will transform the equation into a simpler form, typically a linear or quadratic equation, which we can then solve using standard algebraic techniques. Let’s proceed with the detailed steps to achieve our goal of finding the value(s) of y that satisfy the original equation.

1. Identify the Least Common Denominator (LCD)

The first critical step in solving the equation -4/(y+2) = (1/(5y+10)) + 1 is to identify the least common denominator (LCD). This will allow us to eliminate the fractions and simplify the equation. Observe the denominators: (y+2) and (5y+10). We can factor the second denominator as 5(y+2). This reveals that the LCD is indeed 5(y+2). The LCD is the smallest expression that is divisible by each denominator. Recognizing the LCD is vital because it allows us to multiply both sides of the equation by it, effectively clearing the fractions. This simplification is a key step in solving algebraic equations involving rational expressions. Now that we have identified the LCD, we can proceed to the next step of multiplying both sides of the equation by it.

2. Multiply Both Sides by the LCD

Now that we've identified the LCD as 5(y+2), our next step in solving -4/(y+2) = (1/(5y+10)) + 1 is to multiply both sides of the equation by this LCD. This action is crucial as it eliminates the fractions, transforming the equation into a more manageable form. When multiplying, ensure the LCD is distributed correctly to each term on both sides. For the left side, multiplying 5(y+2) by -4/(y+2) will cancel out the (y+2) term. On the right side, the LCD will multiply both 1/(5y+10) and 1. This step requires careful attention to detail to ensure the terms are correctly multiplied and simplified. After this multiplication, the equation will be free of fractions, making it easier to solve for y. This is a fundamental technique in solving rational equations, and mastering it is essential for more advanced algebraic problems.

3. Simplify the Equation

Following the multiplication by the LCD, the next crucial step in solving -4/(y+2) = (1/(5y+10)) + 1 is to simplify the resulting equation. After multiplying both sides by 5(y+2), we obtain: 5(y+2) * [-4/(y+2)] = 5(y+2) * [1/(5y+10) + 1]. Simplifying this, we get -20 = 1 + 5(y+2). Now, expand the right side: -20 = 1 + 5y + 10. Combine the constants on the right side: -20 = 5y + 11. The simplification process involves basic arithmetic operations and the distributive property. By carefully performing these operations, we reduce the equation to a simpler linear form. This simplified equation is much easier to solve for y, bringing us closer to our final solution. Attention to detail in this step is vital to ensure accuracy in the subsequent steps.

4. Isolate the Variable y

After simplifying the equation, the next key step in solving -4/(y+2) = (1/(5y+10)) + 1 is to isolate the variable y. From the previous step, we have the simplified equation -20 = 5y + 11. To isolate y, we need to move all other terms to the other side of the equation. First, subtract 11 from both sides: -20 - 11 = 5y, which gives us -31 = 5y. Now, to completely isolate y, divide both sides by 5: y = -31/5. This process of isolating the variable involves using inverse operations to undo the operations that are applied to y. By performing these steps carefully, we can determine the value of y that satisfies the equation. Isolating the variable is a fundamental skill in algebra, essential for solving various types of equations.

5. Check for Extraneous Solutions

Once we have a potential solution for y, the final crucial step in solving -4/(y+2) = (1/(5y+10)) + 1 is to check for extraneous solutions. Extraneous solutions are values that satisfy the transformed equation but not the original equation. This often happens when dealing with rational expressions, as certain values can make the denominator zero, rendering the expression undefined. In our original equation, y cannot be -2, as this would make the denominators (y+2) and (5y+10) equal to zero. Our solution, y = -31/5, is not equal to -2, so it is a valid solution. To be completely sure, substitute y = -31/5 back into the original equation to verify that it holds true. This checking process is vital to ensure the accuracy of our solution and to avoid including values that are not valid. Always remember to check for extraneous solutions when solving rational equations.

The Solution

After following the steps meticulously, we've arrived at the solution for the equation -4/(y+2) = (1/(5y+10)) + 1. We identified the LCD, multiplied both sides of the equation by it, simplified the equation, isolated the variable y, and checked for extraneous solutions. The solution we found is y = -31/5. This value does not make any of the denominators in the original equation equal to zero, so it is a valid solution. To ensure complete accuracy, you can substitute this value back into the original equation and verify that both sides are equal. Understanding the process of solving rational equations is a valuable skill in algebra, and this step-by-step guide provides a clear and thorough approach to tackling such problems. Always remember to check your solution to avoid extraneous results.

Common Mistakes to Avoid

When solving equations like -4/(y+2) = (1/(5y+10)) + 1, it's crucial to be aware of common mistakes that can lead to incorrect solutions. One frequent error is failing to distribute correctly when multiplying by the LCD. Ensure that each term on both sides of the equation is multiplied by the LCD. Another mistake is overlooking the potential for extraneous solutions. Always check your final answer by substituting it back into the original equation, especially when dealing with rational expressions where denominators can become zero. Additionally, errors in basic arithmetic, such as adding or subtracting numbers, can derail the entire process. Double-check each step to ensure accuracy. Lastly, not simplifying the equation properly after multiplying by the LCD can lead to a more complex equation that is harder to solve. By being mindful of these common pitfalls, you can increase your accuracy and confidence in solving algebraic equations.

Conclusion

In conclusion, solving the equation -4/(y+2) = (1/(5y+10)) + 1 involves a systematic approach that includes identifying the LCD, multiplying both sides by it, simplifying the equation, isolating the variable, and checking for extraneous solutions. The correct solution to this equation is y = -31/5. This step-by-step guide has provided a detailed explanation of each stage in the process, aiming to equip you with the knowledge and skills to tackle similar algebraic problems. Remember to pay close attention to detail, avoid common mistakes, and always verify your solution to ensure accuracy. Mastering these techniques is essential for success in algebra and related fields. By consistently practicing and applying these methods, you can confidently solve a wide range of equations and enhance your problem-solving abilities.