Solving 1/(c-3) - 1/c = 3/(c(c-3)) A Comprehensive Guide

In the captivating realm of mathematics, rational equations stand as intriguing puzzles, challenging us to navigate the complexities of fractions and algebraic manipulations. These equations, characterized by variables nestled within the denominators of fractions, demand a meticulous approach to unravel their solutions. Our focus today lies on dissecting the rational equation $\frac{1}{c-3}-\frac{1}{c}=\frac{3}{c(c-3)}$, a seemingly intricate expression that holds a hidden solution waiting to be discovered. We embark on a journey to decipher this equation, employing a blend of algebraic techniques and logical reasoning to unveil its true nature. Understanding rational equations is crucial not only for academic pursuits but also for various real-world applications where mathematical models involve fractional relationships. This exploration will provide a comprehensive understanding of the steps involved in solving rational equations, emphasizing the importance of identifying extraneous solutions and ensuring the validity of our results. The process begins with a clear identification of the equation's domain, recognizing any values of the variable that would render the denominators zero, thus making the equation undefined. By systematically eliminating fractions, simplifying expressions, and solving the resulting equation, we can arrive at potential solutions. However, the journey doesn't end there; each potential solution must be meticulously checked against the original equation to rule out extraneous solutions—values that satisfy the transformed equation but not the original. Through this detailed analysis, we gain not only the solution to the specific equation at hand but also a deeper appreciation for the nuances of rational equations and their significance in mathematical problem-solving. Let's delve into the heart of the equation, armed with algebraic tools and a keen eye for detail, to unveil the solution that lies within.

Step 1: Identifying the Forbidden Territory - Determining the Domain

The first crucial step in solving any rational equation is to identify the domain, the set of all permissible values for the variable. This involves pinpointing any values that would make the denominators of the fractions equal to zero, as division by zero is undefined in the mathematical universe. In our equation, $\frac{1}{c-3}-\frac{1}{c}=\frac{3}{c(c-3)}$, we encounter two denominators that demand our attention: c - 3 and c. Setting each of these denominators to zero, we find the forbidden values: c - 3 = 0 implies c = 3, and c = 0. These values, c = 0 and c = 3, are the mathematical landmines we must avoid. They represent points where the equation becomes undefined, and any solution that leads us to these values is deemed extraneous. The domain, therefore, encompasses all real numbers except 0 and 3. This initial step is not merely a formality; it is a fundamental safeguard against erroneous conclusions. By explicitly stating the domain, we set the stage for a rigorous solution process, ensuring that any potential solutions are scrutinized against these forbidden values. This careful consideration of the domain is a hallmark of sound mathematical practice, preventing us from accepting solutions that are mathematically invalid. Identifying the domain is the cornerstone of solving rational equations, providing a framework within which we can confidently manipulate the equation and interpret the results. It's a reminder that mathematics is not just about finding answers; it's about understanding the conditions under which those answers are valid. With the domain clearly established, we proceed to the next stage of our journey, where we will skillfully maneuver the equation to isolate the variable and unveil its true value.

Step 2: Clearing the Fractions - Multiplying by the Least Common Denominator

Having established the domain, our next objective is to eliminate the fractions that cloud the equation. To achieve this, we employ the technique of multiplying both sides of the equation by the least common denominator (LCD). The LCD is the smallest expression that is divisible by all the denominators in the equation. In our case, the denominators are c - 3, c, and c(c - 3). The LCD, therefore, is c(c - 3). Multiplying both sides of the equation $\frac{1}{c-3}-\frac{1}{c}=\frac{3}{c(c-3)}$ by c(c - 3), we embark on a transformative step. On the left-hand side, the multiplication distributes across the two terms, effectively canceling out the denominators. The first term, $\frac{1}{c-3}$, when multiplied by c(c - 3), simplifies to c. The second term, -$\frac{1}{c}$, when multiplied by c(c - 3), simplifies to -(c - 3). On the right-hand side, the multiplication by c(c - 3) neatly cancels out the denominator, leaving us with just 3. The equation now stands transformed, devoid of fractions, in the form c - (c - 3) = 3. This step of clearing fractions is a pivotal maneuver in solving rational equations. It converts the equation into a more manageable form, often a linear or quadratic equation, which we can then solve using standard algebraic techniques. By eliminating the fractions, we simplify the equation, making it easier to isolate the variable and determine its value. This process not only simplifies the equation but also sets the stage for the subsequent steps in our journey, leading us closer to the solution. The transformation from a rational equation to a simpler algebraic form is a testament to the power of algebraic manipulation in unraveling mathematical puzzles. With the fractions cleared, we proceed to simplify the equation further, paving the way for the final unveiling of the solution.

Step 3: Simplifying the Equation - Unveiling the Hidden Structure

With the fractions successfully cleared, the equation c - (c - 3) = 3 stands before us, poised for simplification. This step is crucial in unveiling the underlying structure of the equation, making it easier to isolate the variable and arrive at a solution. We begin by carefully distributing the negative sign in the expression c - (c - 3). This yields c - c + 3, where the subtraction of (c - 3) becomes the addition of its negative. The equation now reads c - c + 3 = 3. A keen observation reveals that c and -c are additive inverses, canceling each other out. This simplification leaves us with the elegant equation 3 = 3. This equation, while seemingly trivial, holds a profound implication. It states that the equation is true for all values of c that are within the domain. The simplification process has stripped away the complexities of the original equation, revealing a fundamental truth: the equation is an identity. Simplifying the equation is a critical step in the problem-solving process. It not only reduces the complexity of the equation but also unveils its true nature. In this case, the simplification has revealed that the equation is not a conditional equation, one that is true only for specific values of the variable, but an identity, one that is true for all values within the domain. This realization is a significant milestone in our journey, guiding our understanding of the solution set. The equation 3 = 3 is a testament to the power of simplification in revealing hidden mathematical truths. It demonstrates how algebraic manipulations can strip away the superficial complexities of an equation, exposing its core essence. With the equation simplified, we are now poised to interpret the results and draw conclusions about the solution.

Step 4: Interpreting the Solution - A Deeper Understanding

The simplified equation, 3 = 3, presents a unique scenario. It's an identity, a statement that holds true regardless of the value of c. However, we must remember the domain we established in Step 1: c cannot be 0 or 3, as these values would make the original equation undefined. This is a critical juncture where we must reconcile the algebraic result with the constraints imposed by the domain. The equation 3 = 3 suggests that any value of c would satisfy the equation. But the domain restrictions remind us that c cannot be 0 or 3. This means that while the equation holds true for all other values, these two values are excluded from the solution set. This leads us to a profound realization: despite the algebraic simplification suggesting otherwise, there is no solution that satisfies the original equation within its defined domain. This is a classic example of an extraneous solution, a value that satisfies a transformed equation but not the original. Extraneous solutions often arise in rational equations due to the process of clearing denominators, which can introduce solutions that do not align with the original equation's restrictions. Interpreting the solution in the context of the domain is a vital step in solving rational equations. It's a reminder that mathematical solutions are not just about algebraic manipulations; they are about understanding the conditions under which those manipulations are valid. The presence of extraneous solutions underscores the importance of checking potential solutions against the original equation and the domain. In this case, the equation 3 = 3 might lead one to believe that any value of c works, but the domain restrictions reveal the truth: there is no solution. This nuanced understanding of the solution set is a hallmark of mathematical rigor, ensuring that our conclusions are not only algebraically sound but also logically consistent with the original problem.

Conclusion: The Absence of a Solution

In our meticulous exploration of the rational equation $\frac{1}{c-3}-\frac{1}{c}=\frac{3}{c(c-3)}$, we've journeyed through the essential steps of solving such equations. We began by identifying the domain, recognizing the forbidden values of c that would render the equation undefined. We then cleared the fractions by multiplying both sides by the least common denominator, transforming the equation into a simpler form. Simplification led us to the intriguing equation 3 = 3, an identity that holds true regardless of the value of c. However, it was the critical step of interpreting the solution within the context of the domain that revealed the true nature of the problem. The domain restrictions, c ≠ 0 and c ≠ 3, clashed with the implication of the identity, leading us to the conclusion that there is no solution to the original equation. This journey underscores the importance of a comprehensive approach to solving rational equations. It's not enough to simply manipulate the equation algebraically; we must also be mindful of the domain and the potential for extraneous solutions. The absence of a solution in this case is not a failure but a testament to the rigor of mathematical analysis. We have successfully navigated the complexities of the equation, revealing its true nature. The process of solving rational equations is not just about finding numbers; it's about understanding the interplay between algebra, logic, and the constraints imposed by the problem itself. This understanding is a valuable asset, not only in mathematics but also in various fields where mathematical models play a crucial role. The equation $\frac{1}{c-3}-\frac{1}{c}=\frac{3}{c(c-3)}$ serves as a compelling example of how a seemingly straightforward equation can harbor subtle complexities, demanding a careful and methodical approach to uncover the truth. The final answer is D. no solution.