Solving The Rational Equation Y/(y-4) - 4/(y+4) = 32/(y^2-16)

Introduction: Navigating the Realm of Rational Equations

In the captivating world of mathematics, equations often present themselves as intriguing puzzles, challenging us to unravel their hidden solutions. Among these mathematical enigmas, rational equations hold a special allure. These equations, characterized by fractions with variables in their denominators, demand a meticulous approach and a keen eye for algebraic manipulation. This article delves into the fascinating realm of rational equations, focusing on the specific equation y/(y-4) - 4/(y+4) = 32/(y^2-16). We will embark on a step-by-step journey to decipher its solution, shedding light on the underlying principles and techniques involved. So, let's immerse ourselves in the realm of mathematical exploration and uncover the secrets held within this equation.

Deconstructing the Equation: A Foundation for Success

Before we embark on the journey of solving this equation, it's crucial to first understand its structure. Our focus equation is y/(y-4) - 4/(y+4) = 32/(y^2-16), a rational equation. To solve it effectively, we need to simplify it by eliminating the fractions. The first step in this process is to identify the common denominator of all the fractions in the equation. This involves factoring the denominators and finding the least common multiple (LCM). In our equation, the denominators are y-4, y+4, and y^2-16. Notice that y^2-16 can be factored as (y-4)(y+4), using the difference of squares identity. This insightful observation reveals that the common denominator for our equation is (y-4)(y+4). By recognizing this, we've laid the foundation for a smoother solution process.

The Art of Elimination: Multiplying by the Common Denominator

With the common denominator identified as (y-4)(y+4), we can now proceed to eliminate the fractions from the equation. This is achieved by multiplying both sides of the equation by the common denominator. When we multiply each term of the equation y/(y-4) - 4/(y+4) = 32/(y^2-16) by (y-4)(y+4), a remarkable transformation occurs. The denominators gracefully cancel out, leaving us with a simplified equation devoid of fractions. Let's perform this multiplication step by step. Multiplying the first term, y/(y-4), by (y-4)(y+4), the (y-4) terms cancel out, leaving us with y(y+4). For the second term, -4/(y+4), the (y+4) terms cancel, resulting in -4(y-4). On the right side of the equation, 32/(y^2-16), the entire denominator (y^2-16), which is (y-4)(y+4), cancels out, leaving us with just 32. Thus, the equation transforms into y(y+4) - 4(y-4) = 32. This crucial step has effectively eliminated the fractions, paving the way for a more manageable algebraic equation.

Unveiling the Quadratic: Expanding and Simplifying

Now that we've eliminated the fractions, our equation y(y+4) - 4(y-4) = 32 takes on a more familiar form. To further simplify it, we need to expand the terms and combine like terms. Let's begin by expanding the products. Multiplying y by (y+4) gives us y^2 + 4y. Next, multiplying -4 by (y-4) results in -4y + 16. So, our equation now becomes y^2 + 4y - 4y + 16 = 32. Notice that the 4y and -4y terms cancel each other out, simplifying the equation further. We are left with y^2 + 16 = 32. To solve for y, we need to isolate the y^2 term. Subtracting 16 from both sides of the equation gives us y^2 = 16. This is a significant milestone, as we've now transformed the equation into a quadratic form, which we can readily solve.

Extracting the Roots: Solving the Quadratic Equation

With our equation simplified to y^2 = 16, we are now poised to extract the roots and find the values of y that satisfy the equation. To solve for y, we take the square root of both sides of the equation. Remember that when taking the square root, we must consider both the positive and negative roots. The square root of 16 is both 4 and -4. Therefore, the solutions to the equation y^2 = 16 are y = 4 and y = -4. These values represent the potential solutions to our original rational equation. However, it's crucial to remember that in rational equations, we must always check for extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation.

The Check for Extraneous Solutions: A Crucial Verification Step

In the realm of rational equations, the check for extraneous solutions is a critical step that cannot be overlooked. Extraneous solutions are values that emerge as solutions during the solving process but do not satisfy the original equation. This often happens when we perform operations that can introduce solutions that were not initially present, such as multiplying both sides of the equation by an expression containing a variable. To check for extraneous solutions, we must substitute each potential solution back into the original equation and verify that it holds true. Our original equation is y/(y-4) - 4/(y+4) = 32/(y^2-16), and our potential solutions are y = 4 and y = -4. Let's start by substituting y = 4 into the original equation. We get 4/(4-4) - 4/(4+4) = 32/(4^2-16). This simplifies to 4/0 - 4/8 = 32/0. Notice that we have division by zero in the first term and on the right side of the equation. Division by zero is undefined in mathematics, which means that y = 4 is an extraneous solution. Now, let's substitute y = -4 into the original equation. We get -4/(-4-4) - 4/(-4+4) = 32/((-4)^2-16). This simplifies to -4/-8 - 4/0 = 32/0. Again, we encounter division by zero in the second term and on the right side of the equation. This indicates that y = -4 is also an extraneous solution. Therefore, after checking for extraneous solutions, we find that neither y = 4 nor y = -4 is a valid solution to the original equation. This leads us to the conclusion that the equation has no solution.

Conclusion: The Absence of a Solution and the Importance of Verification

In our comprehensive exploration of the equation y/(y-4) - 4/(y+4) = 32/(y^2-16), we embarked on a journey of algebraic manipulation, simplification, and verification. We meticulously followed the steps involved in solving rational equations, including identifying the common denominator, eliminating fractions, expanding terms, and solving the resulting quadratic equation. Our efforts led us to two potential solutions: y = 4 and y = -4. However, the crucial step of checking for extraneous solutions revealed a surprising twist. Both potential solutions resulted in division by zero when substituted back into the original equation, rendering them invalid. This unexpected outcome leads us to the final conclusion: the equation y/(y-4) - 4/(y+4) = 32/(y^2-16) has no solution. This journey underscores the importance of the verification step in solving rational equations. It serves as a reminder that not all solutions obtained through algebraic manipulation are necessarily valid solutions to the original equation. The check for extraneous solutions acts as a safeguard, ensuring that we arrive at accurate and meaningful conclusions in our mathematical endeavors.