Solving Rational Equations A Step-by-Step Guide For -2/(x-3) = X/(x-6)
In the realm of mathematics, solving equations is a fundamental skill, and when dealing with rational equations, the process involves a few crucial steps. This article delves into a detailed solution for the equation -2/(x-3) = x/(x-6), providing a step-by-step guide to finding all possible solutions. We will explore the underlying concepts, potential pitfalls, and verification methods to ensure a thorough understanding of the solution process. Understanding how to solve rational equations is crucial in various fields, including physics, engineering, and economics, where these types of equations often model real-world scenarios. The ability to manipulate algebraic expressions and solve for unknown variables is a cornerstone of mathematical literacy.
When presented with an equation like -2/(x-3) = x/(x-6), it is essential to approach it systematically. First, we must identify the restrictions on the variable x. Since division by zero is undefined, we need to exclude any values of x that would make the denominators equal to zero. In this case, x cannot be 3 or 6. These values are critical to note, as they will help us determine if our final solutions are valid. The next step involves clearing the fractions by multiplying both sides of the equation by the least common denominator (LCD). This simplifies the equation into a more manageable form, typically a polynomial equation. The LCD in this case is the product of the denominators, (x-3)(x-6). Multiplying both sides by this LCD is a key step in transforming the rational equation into a quadratic equation, which we can then solve using standard methods. By following this structured approach, we can confidently find the solutions to the equation and avoid common mistakes.
Once we have cleared the fractions and simplified the equation, we will likely end up with a quadratic equation. Quadratic equations can be solved using several methods, including factoring, completing the square, or using the quadratic formula. In this instance, we will demonstrate the factoring method, as it often provides a straightforward solution when the quadratic expression can be easily factored. After obtaining the potential solutions, we must always verify them against the original equation, especially considering the initial restrictions we identified. This verification step ensures that we are only accepting solutions that are valid within the domain of the equation. This process of solving rational equations not only reinforces algebraic skills but also highlights the importance of precision and attention to detail in mathematical problem-solving. The ability to accurately solve these equations is an invaluable asset in various academic and professional contexts. By carefully considering each step, from identifying restrictions to verifying solutions, we can confidently tackle even the most complex rational equations.
Step-by-Step Solution of -2/(x-3) = x/(x-6)
To solve the given rational equation, -2/(x-3) = x/(x-6), we will follow a methodical approach, ensuring we identify and address any restrictions along the way. This systematic method will help us find the correct solutions while avoiding common pitfalls in algebraic manipulations. The equation involves fractions with variables in the denominator, so it's crucial to proceed carefully to prevent errors and to ensure our solutions are valid.
The first critical step is to identify the values of x that would make the denominators zero, as these values are not permissible. For the denominator (x-3), x cannot be 3, and for the denominator (x-6), x cannot be 6. These restrictions are essential to note because any solution we find must not coincide with these values. If a potential solution does equal 3 or 6, it must be discarded as it would make the original equation undefined. These excluded values are a fundamental aspect of solving rational equations, and neglecting them can lead to incorrect answers. By clearly stating these restrictions upfront, we set the stage for a more accurate and reliable solution process.
Next, we clear the fractions by multiplying both sides of the equation by the least common denominator (LCD). The LCD for this equation is (x-3)(x-6). Multiplying both sides by this expression eliminates the fractions, transforming the equation into a more manageable form. This step is crucial in simplifying the equation and allowing us to proceed with standard algebraic techniques. When we multiply the left side, -2/(x-3), by (x-3)(x-6), the (x-3) terms cancel out, leaving us with -2(x-6). Similarly, when we multiply the right side, x/(x-6), by (x-3)(x-6), the (x-6) terms cancel out, leaving us with x(x-3). This process converts the rational equation into a simpler equation that we can solve by expanding and rearranging terms. The elimination of fractions is a pivotal step in making the equation solvable and sets the stage for finding the potential solutions.
Now, we expand and simplify the equation. Distributing the -2 on the left side gives us -2x + 12, and distributing the x on the right side gives us x^2 - 3x. So the equation becomes -2x + 12 = x^2 - 3x. To solve for x, we need to set the equation to zero, which means rearranging the terms to one side. We can do this by adding 2x and subtracting 12 from both sides, resulting in 0 = x^2 - 5x - 12. This quadratic equation is now in the standard form, which allows us to solve it using factoring, completing the square, or the quadratic formula. Factoring is often the most straightforward method when the quadratic expression can be easily factored. The transformation of the rational equation into this standard quadratic form is a significant milestone in the solution process, paving the way for identifying the values of x that satisfy the original equation.
To solve the quadratic equation x^2 - 5x - 12 = 0, we look for two numbers that multiply to -12 and add up to -5. In this case, the equation can be factored as (x-4)(x-1) = 0. Setting each factor equal to zero gives us two potential solutions: x - 4 = 0, which implies x = 4, and x - 1 = 0, which implies x = 1. These are the values of x that satisfy the quadratic equation, but we must now verify these solutions against the original rational equation and the restrictions we identified earlier. The process of factoring and finding potential solutions is a key step in solving quadratic equations, and it's essential to perform this step accurately. However, the job is not yet complete; we must now ensure that these solutions are valid within the context of the original problem.
Finally, we verify the solutions. We have two potential solutions, x = 4 and x = 1. We need to check if these values satisfy the original equation, -2/(x-3) = x/(x-6), and if they violate our initial restrictions (x ≠3 and x ≠6). For x = 4, substituting into the equation gives us -2/(4-3) = -2/1 = -2 on the left side, and 4/(4-6) = 4/(-2) = -2 on the right side. Thus, x = 4 is a valid solution. For x = 1, substituting into the equation gives us -2/(1-3) = -2/(-2) = 1 on the left side, and 1/(1-6) = 1/(-5) = -1/5 on the right side. Since 1 ≠-1/5, x = 1 is not a valid solution. Therefore, the only solution to the original equation is x = 4. This verification step is crucial in solving rational equations, as it ensures that we discard any extraneous solutions that may arise during the algebraic manipulation process. By thoroughly verifying our solutions, we can confidently arrive at the correct answer.
Conclusion: Mastering the Art of Solving Rational Equations
In conclusion, solving the rational equation -2/(x-3) = x/(x-6) requires a systematic approach that includes identifying restrictions, clearing fractions, simplifying the equation, finding potential solutions, and crucially, verifying those solutions. This comprehensive process ensures that we arrive at the correct answer and avoid extraneous solutions. The steps involved highlight the importance of attention to detail and precision in algebraic manipulations. Solving rational equations is a valuable skill in mathematics, with applications in various fields, reinforcing the significance of mastering this technique.
We began by recognizing the importance of identifying restrictions on the variable x, as division by zero is undefined. This step is fundamental to ensuring the validity of our solutions. By noting that x cannot be 3 or 6, we set the stage for a more accurate solution process. The subsequent step involved clearing the fractions by multiplying both sides of the equation by the least common denominator, (x-3)(x-6). This transformed the equation into a more manageable form, allowing us to proceed with standard algebraic methods. The expansion and simplification of the equation led us to a quadratic equation, which we then solved by factoring. The potential solutions obtained were x = 4 and x = 1. However, the job was not complete until we verified these solutions against the original equation and the initial restrictions.
The verification process revealed that only x = 4 is a valid solution. Substituting x = 4 into the original equation demonstrated that it satisfies the equation. On the other hand, substituting x = 1 showed that it does not satisfy the equation, making it an extraneous solution. This step underscores the critical importance of verification in solving rational equations. Extraneous solutions can arise due to the algebraic manipulations involved, and it is only through verification that we can identify and discard them. The final solution, x = 4, is the only value that satisfies the given equation without violating any restrictions. This thorough approach to solving rational equations not only provides the correct answer but also reinforces the importance of careful and methodical problem-solving in mathematics. The ability to accurately solve these types of equations is a valuable asset in various academic and professional settings.
By mastering the art of solving rational equations, we gain a deeper understanding of algebraic principles and their applications. The systematic approach outlined in this article can be applied to a wide range of similar problems, reinforcing the importance of a structured and methodical approach in mathematics. The steps involved in solving rational equations—identifying restrictions, clearing fractions, simplifying, solving, and verifying—are applicable to many other areas of mathematics, making this a fundamental skill to develop. The ability to confidently tackle these equations demonstrates a strong foundation in algebraic manipulation and problem-solving, which are essential for success in advanced mathematical studies and various real-world applications. This comprehensive guide aims to equip readers with the knowledge and skills necessary to approach rational equations with confidence and accuracy.
Correct Answer:
F. 4
