Solving The Rational Equation -8/(x-6) = 10/x A Step-by-Step Guide

In this article, we will delve into the process of solving the equation -8/(x-6) = 10/x. This equation falls under the category of rational equations, which involve fractions with variables in the denominator. Solving such equations requires careful manipulation to eliminate the fractions and isolate the variable x. We will explore the steps involved, including finding a common denominator, cross-multiplication, and solving the resulting algebraic equation. Understanding how to solve rational equations is crucial in various fields of mathematics and its applications, such as physics, engineering, and economics. This article aims to provide a comprehensive guide, suitable for students and anyone interested in enhancing their algebraic skills. We will not only present the solution but also explain the underlying concepts and potential pitfalls to avoid. So, let's embark on this journey of solving rational equations and strengthen our mathematical prowess.

Rational equations, like the one we are tackling, -8/(x-6) = 10/x, are a fascinating area of algebra. To fully grasp how to solve them, it's essential to first understand what they are and the challenges they present. A rational equation is essentially an equation that contains one or more fractions where the numerator and/or the denominator are polynomials. In our case, both sides of the equation are fractions with expressions involving x. The primary hurdle in solving these equations is the presence of variables in the denominator. This means we can't directly apply standard algebraic techniques like simple addition or subtraction until we've eliminated the fractions. One common strategy is to find a common denominator and then multiply both sides of the equation by it. This eliminates the fractions, transforming the rational equation into a more manageable algebraic form, often a linear or quadratic equation. However, it's crucial to be mindful of potential extraneous solutions. These are solutions that satisfy the transformed equation but not the original rational equation. This typically happens when a solution makes the denominator of the original equation zero, which is undefined in mathematics. Therefore, after finding the solutions, we must always check them against the original equation to ensure they are valid. This step is a critical part of the process and ensures the accuracy of our solution. Ignoring it can lead to incorrect answers and a misunderstanding of the problem. Solving rational equations is a fundamental skill in algebra, and mastering it opens doors to solving more complex problems in various fields.

Let's break down the solution to the equation -8/(x-6) = 10/x step-by-step. This will provide a clear understanding of the process and the reasoning behind each action.

1. Identify the Equation: The equation we are working with is -8/(x-6) = 10/x. This is a rational equation because it contains fractions with x in the denominators.

2. Eliminate the Fractions: To get rid of the fractions, we need to find a common denominator. In this case, the common denominator is the product of the two denominators, which is x(x-6). We will multiply both sides of the equation by this common denominator. This gives us: x(x-6) * [-8/(x-6)] = x(x-6) * [10/x].

3. Simplify the Equation: On the left side, the (x-6) terms cancel out, leaving us with -8x. On the right side, the x terms cancel out, leaving us with 10*(x-6). So, the equation now looks like this: -8x = 10(x-6).

4. Distribute: We need to distribute the 10 on the right side of the equation: -8x = 10x - 60.

5. Combine Like Terms: Now, let's get all the x terms on one side and the constants on the other. Add 8x to both sides: 0 = 18x - 60.

6. Isolate the Variable: Add 60 to both sides: 60 = 18x.

7. Solve for x: Divide both sides by 18 to isolate x: x = 60/18. We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 6. So, x = 10/3.

8. Check for Extraneous Solutions: This is a critical step. We need to plug x = 10/3 back into the original equation to make sure it doesn't make any of the denominators zero. In the original equation, the denominators are (x-6) and x. If x = 10/3, then (x-6) = 10/3 - 6 = -8/3, which is not zero. Also, x = 10/3 is not zero. Therefore, x = 10/3 is a valid solution.

9. Final Solution: The solution to the equation -8/(x-6) = 10/x is x = 10/3.

By following these steps carefully, we have successfully solved the rational equation. Remember, the key is to eliminate the fractions, simplify the equation, and always check for extraneous solutions.

Solving rational equations can be tricky, and there are several potential pitfalls that students often encounter. Being aware of these common mistakes and learning how to avoid them is crucial for mastering this type of equation. One of the most frequent errors is forgetting to check for extraneous solutions. As we discussed earlier, these are solutions that arise during the solving process but do not satisfy the original equation. They typically occur when a solution makes the denominator of the original equation zero, resulting in an undefined expression. To avoid this, always substitute your solutions back into the original equation and verify that they do not lead to any undefined terms. Another common mistake is incorrectly applying the distributive property. When multiplying both sides of the equation by the common denominator, it's essential to distribute the multiplication to every term correctly. A failure to do so can lead to an incorrect simplification and ultimately, a wrong solution. Take extra care when distributing, especially if there are multiple terms or negative signs involved. Another pitfall is not finding the least common denominator (LCD) properly. While any common denominator will work, using the LCD simplifies the equation more efficiently. Incorrectly identifying the LCD can lead to larger numbers and more complex calculations, increasing the chance of making an error. Always double-check your LCD to ensure it's the smallest expression that all denominators divide into evenly. Finally, mistakes in basic algebra, such as combining like terms or performing arithmetic operations, can also derail the solution. It's crucial to be meticulous and double-check each step to avoid these errors. Practice and attention to detail are key to successfully navigating the challenges of solving rational equations. By being aware of these potential pitfalls and taking steps to avoid them, you can significantly improve your accuracy and confidence in solving these equations. Always remember to check your work, and don't hesitate to break down the problem into smaller, more manageable steps.

Rational equations are not just abstract mathematical concepts; they have numerous real-world applications in various fields. Understanding these applications can help to appreciate the practical value of learning how to solve them. In physics, rational equations are often used to describe relationships between quantities such as velocity, distance, and time, especially in situations involving motion with varying speeds. For example, calculating the average speed of a journey with different segments traveled at different speeds can involve solving a rational equation. In engineering, these equations are crucial in designing electrical circuits, where they help to determine the relationships between resistance, voltage, and current. They are also used in fluid dynamics to model flow rates and pressures in pipes and channels. In economics, rational equations can be used to model supply and demand curves, cost-benefit analyses, and other economic relationships. For instance, the break-even point for a business, where total revenue equals total costs, can be found using a rational equation. In chemistry, these equations are applied in chemical kinetics to describe reaction rates and equilibrium constants. The relationships between concentrations of reactants and products at equilibrium often involve rational expressions. Even in everyday situations, rational equations can be used. For example, calculating the time it takes to complete a task when multiple people are working on it at different rates involves a rational equation. Similarly, calculating the cost per item when buying in bulk with discounts can also be modeled using rational equations. These examples illustrate that rational equations are a powerful tool for modeling and solving a wide range of real-world problems. By mastering the techniques for solving these equations, you not only enhance your mathematical skills but also gain a valuable tool for problem-solving in various fields. The ability to recognize and apply rational equations in practical situations is a valuable asset in many professions and everyday life.

In this comprehensive guide, we have thoroughly explored the process of solving the rational equation -8/(x-6) = 10/x. We began by defining rational equations and understanding the challenges they present, particularly the presence of variables in the denominators and the potential for extraneous solutions. We then meticulously worked through the step-by-step solution, which involved eliminating fractions, simplifying the equation, isolating the variable, and most importantly, checking for extraneous solutions. We emphasized the critical nature of this last step to ensure the validity of our solution. Furthermore, we discussed common pitfalls that students often encounter when solving rational equations, such as forgetting to check for extraneous solutions, incorrectly applying the distributive property, and making mistakes in basic algebra. By highlighting these potential errors and providing strategies to avoid them, we aimed to equip you with the tools to approach these equations with confidence and accuracy. Beyond the mechanics of solving, we also explored the real-world applications of rational equations across various disciplines, including physics, engineering, economics, and chemistry. These examples demonstrated the practical significance of mastering this mathematical concept and its relevance to solving real-world problems. In conclusion, solving rational equations is a fundamental skill in algebra with wide-ranging applications. By understanding the underlying principles, following a systematic approach, and being mindful of potential pitfalls, you can successfully tackle these equations and enhance your problem-solving abilities. We encourage you to continue practicing and exploring the fascinating world of mathematics.