Solving Y/(y-4) - 4/(y+4) = 32/(y^2-16) A Step-by-Step Guide

In the realm of algebra, solving equations is a fundamental skill. This article delves into the step-by-step solution of a rational equation, providing a comprehensive guide for students and enthusiasts alike. We will tackle the equation ${\frac{y}{y-4}-\frac{4}{y+4}=\frac{32}{y^2-16}}$, which involves fractions with polynomial expressions. Understanding how to solve such equations is crucial for various applications in mathematics and other scientific disciplines. Our approach will focus on clarity and detail, ensuring that each step is thoroughly explained. This method not only helps in finding the solution but also enhances the understanding of the underlying algebraic principles. Let’s embark on this mathematical journey, breaking down the complexities and revealing the elegant solution to this equation.

Understanding Rational Equations

Rational equations, which form the core of our discussion, are equations that contain rational expressions—expressions that are essentially fractions with polynomials in the numerator and denominator. Solving rational equations requires a systematic approach, as we need to handle fractions and potential extraneous solutions. The first critical step in solving these equations is identifying the domain, which means determining the values of the variable that do not make the denominator zero. These values are excluded from the solution set because division by zero is undefined. Once the domain is established, we proceed to clear the fractions. This is typically done by multiplying both sides of the equation by the least common denominator (LCD). The LCD is the smallest expression that is divisible by each denominator in the equation. Clearing fractions simplifies the equation, transforming it into a more manageable form, often a polynomial equation. However, it's crucial to remember that multiplying by an expression containing a variable can introduce extraneous solutions—solutions that satisfy the transformed equation but not the original one. Therefore, after finding the solutions to the simplified equation, we must check each solution against the original equation to ensure it is valid. This process of checking for extraneous solutions is a vital step in solving rational equations, ensuring the accuracy of our final answer.

Step 1: Identifying the Domain

The initial step in solving the equation ${\frac{y}{y-4}-\frac{4}{y+4}=\frac{32}{y^2-16}}$ is to pinpoint the domain of the variable y. The domain encompasses all possible values that y can take without rendering the equation undefined. In the context of rational equations, undefined scenarios arise when the denominator of any fraction equates to zero. Consequently, our focus shifts to identifying the values of y that nullify the denominators in our equation. We have three denominators to consider: ${y - 4}$, ${y + 4}$, and ${y^2 - 16}$. Setting each of these to zero, we can determine the values to exclude from our domain. For ${y - 4 = 0}$, we find that ${y = 4}$ is a value that makes the denominator zero. Similarly, for ${y + 4 = 0}$, we get ${y = -4}$ as another excluded value. The third denominator, ${y^2 - 16}$, can be factored into ${(y - 4)(y + 4)}$. This reveals that setting ${y^2 - 16 = 0}$ also leads to ${y = 4}$ or ${y = -4}$ as solutions, which are the same values we identified earlier. Therefore, the domain of our equation consists of all real numbers except 4 and -4. These exclusions are critical, as any solution we find later must be checked against these values to avoid extraneous solutions. By identifying and excluding these values upfront, we ensure the integrity of our solution process, maintaining mathematical rigor and accuracy.

Step 2: Finding the Least Common Denominator (LCD)

Having established the domain for our equation, the next pivotal step is to determine the least common denominator (LCD). The LCD is the smallest expression that each denominator in the equation can divide into without leaving a remainder. This concept is crucial for clearing the fractions in our equation, thereby simplifying it into a more manageable form. In our equation, ${\frac{y}{y-4}-\frac{4}{y+4}=\frac{32}{y^2-16}}$, we have three denominators: ${y - 4}$, ${y + 4}$, and ${y^2 - 16}$. To find the LCD, we need to consider the factorization of each denominator. The first two denominators, ${y - 4}$ and ${y + 4}$, are already in their simplest form. However, the third denominator, ${y^2 - 16}$, is a difference of squares and can be factored into ${(y - 4)(y + 4)}$. This factorization is key to finding the LCD. The LCD must include each factor that appears in any of the denominators. In this case, we have the factors ${(y - 4)}$ and ${(y + 4)}$. Since ${(y - 4)(y + 4)}$ encompasses both factors, it serves as the LCD for our equation. Identifying the LCD as ${(y - 4)(y + 4)}$ is a critical step, as it allows us to eliminate the fractions by multiplying both sides of the equation by this expression. This process transforms the rational equation into a simpler algebraic equation, paving the way for an easier solution.

Step 3: Multiplying Both Sides by the LCD

With the least common denominator (LCD) identified as ${(y - 4)(y + 4)}$, our next strategic move is to multiply both sides of the equation ${\frac{y}{y-4}-\frac{4}{y+4}=\frac{32}{y^2-16}}$ by this LCD. This action is pivotal as it clears the fractions from the equation, transforming it into a more manageable algebraic form. When we multiply the left side of the equation by ${(y - 4)(y + 4)}$, we distribute this LCD across each term. For the first term, ${\frac{y}{y-4}}$, multiplying by ${(y - 4)(y + 4)}$ results in the ${(y - 4)}$ terms canceling out, leaving us with ${y(y + 4)}$. Similarly, for the second term, ${\frac{4}{y+4}}$, multiplying by ${(y - 4)(y + 4)}$ causes the ${(y + 4)}$ terms to cancel, resulting in ${4(y - 4)}$. On the right side of the equation, we have ${\frac{32}{y^2-16}}$. Recognizing that ${y^2 - 16}$ is equivalent to ${(y - 4)(y + 4)}$, multiplying by the LCD causes the entire denominator to cancel out, leaving us with just 32. The equation now transforms into ${y(y + 4) - 4(y - 4) = 32}$. This step is crucial because it simplifies the original rational equation into a standard polynomial equation. By eliminating the fractions, we make the equation significantly easier to solve, allowing us to proceed with algebraic manipulations to find the value(s) of y that satisfy the equation.

Step 4: Simplifying and Solving the Equation

Having cleared the fractions, we now focus on simplifying and solving the resulting equation, ${y(y + 4) - 4(y - 4) = 32}$. The initial task involves expanding the terms on the left side of the equation. Distributing y in the first term, ${y(y + 4)}$, yields ${y^2 + 4y}$. Similarly, distributing -4 in the second term, ${-4(y - 4)}$, gives us ${-4y + 16}$. Thus, the equation becomes ${y^2 + 4y - 4y + 16 = 32}$. Next, we simplify the equation by combining like terms. We notice that ${4y}$ and ${-4y}$ cancel each other out, leaving us with ${y^2 + 16 = 32}$. To further simplify and isolate the variable, we subtract 16 from both sides of the equation, resulting in ${y^2 = 16}$. Now, to solve for y, we take the square root of both sides of the equation. Remember, when taking the square root, we must consider both the positive and negative roots. Therefore, we get ${y = \pm 4}$, which means y can be either 4 or -4. These values are potential solutions to our equation. However, it's crucial to recall the domain restrictions we identified earlier. The next step is to check these solutions against the original equation to ensure they are valid and not extraneous.

Step 5: Checking for Extraneous Solutions

Having found the potential solutions ${y = 4}$ and ${y = -4}$, the critical next step is to check for extraneous solutions. Extraneous solutions are values that satisfy the simplified equation but not the original rational equation. This often occurs when we clear fractions by multiplying both sides of the equation by an expression that contains the variable. To check for extraneous solutions, we substitute each potential solution back into the original equation, ${\frac{y}{y-4}-\frac{4}{y+4}=\frac{32}{y^2-16}}$, and see if it holds true. First, let's consider ${y = 4}$. Substituting this value into the equation, we immediately encounter a problem. The denominators ${y - 4}$ and ${y^2 - 16}$ become zero, leading to division by zero, which is undefined. Therefore, ${y = 4}$ is an extraneous solution and must be discarded. Next, we check ${y = -4}$. Substituting this value into the equation, the denominator ${y + 4}$ becomes zero, also leading to division by zero. Consequently, ${y = -4}$ is also an extraneous solution and must be rejected. Since both potential solutions are extraneous, we conclude that the original equation has no solution. This underscores the importance of checking for extraneous solutions in rational equations, as it ensures that we only accept valid solutions.

Conclusion: The Absence of a Solution

In conclusion, after a thorough examination of the rational equation ${\frac{y}{y-4}-\frac{4}{y+4}=\frac{32}{y^2-16}}$, we've navigated through the essential steps of solving such equations. We began by identifying the domain, excluding values that would make the denominators zero. We then determined the least common denominator (LCD) to clear the fractions, transforming the equation into a simpler algebraic form. Upon solving the simplified equation, we arrived at two potential solutions: ${y = 4}$ and ${y = -4}$. However, the crucial step of checking for extraneous solutions revealed that both of these values result in division by zero in the original equation, rendering them invalid. Therefore, we conclude that the equation ${\frac{y}{y-4}-\frac{4}{y+4}=\frac{32}{y^2-16}}$ has no solution. This outcome highlights the significance of not only mastering the algebraic manipulations involved in solving equations but also understanding the underlying principles of mathematical validity. The absence of a solution in this case is a testament to the importance of checking for extraneous solutions, ensuring the accuracy and integrity of our mathematical results. This process reinforces the comprehensive approach required for solving rational equations, emphasizing the need for both algebraic skill and careful consideration of domain restrictions.

After meticulously working through the equation ${\frac{y}{y-4}-\frac{4}{y+4}=\frac{32}{y^2-16}}$, we have arrived at a definitive conclusion. By methodically addressing each step—identifying the domain, determining the least common denominator, clearing fractions, solving the resulting equation, and crucially, checking for extraneous solutions—we have found that the equation yields no valid solution. The potential solutions, ${y = 4}$ and ${y = -4}$, were both deemed extraneous due to causing division by zero in the original equation. This comprehensive process underscores the importance of rigorous verification in mathematical problem-solving, particularly when dealing with rational equations. Therefore, the final answer is that there is no solution to the given equation. This conclusion not only resolves the specific problem at hand but also reinforces the broader principle that not all algebraic manipulations lead to valid solutions, and careful checking is paramount.