Solving The Equation 4/(w-1) - 3/(w+2) = 5/w A Step By Step Guide
Introduction
In this article, we will delve into the process of solving the equation 4/(w-1) - 3/(w+2) = 5/w. This equation involves fractions with variables in the denominators, which requires a systematic approach to eliminate the fractions and solve for the unknown variable, w. We will guide you through each step, ensuring a clear understanding of the techniques used in solving such algebraic equations. Our goal is to provide a comprehensive solution, expressing the answers as integers or fractions in their simplest form. This problem falls under the category of algebraic equations, a fundamental topic in mathematics that finds applications in various fields, including physics, engineering, and economics. Mastering the techniques to solve these equations is crucial for anyone pursuing studies or careers in these areas. So, let's embark on this mathematical journey and unravel the solution to this equation.
Understanding the Equation
The given equation is 4/(w-1) - 3/(w+2) = 5/w. This is a rational equation because it involves algebraic fractions. To solve it, our primary goal is to eliminate the denominators. This is achieved by multiplying both sides of the equation by the least common denominator (LCD) of the fractions. The LCD is the smallest expression that is divisible by each denominator. In this case, the denominators are w-1, w+2, and w. Therefore, the LCD is w(w-1)(w+2). By multiplying each term in the equation by this LCD, we will transform the equation into a more manageable form, typically a quadratic equation, which we can then solve using standard methods. Before we proceed, it's essential to note the restrictions on w. The denominators cannot be zero, so w cannot be 1, -2, or 0. These values would make the original equation undefined. Keeping these restrictions in mind is crucial to ensure that our final solutions are valid.
Step-by-Step Solution
To effectively solve this equation, we'll follow a series of carefully structured steps. This process involves clearing the fractions, simplifying the resulting polynomial, and then finding the values of w that satisfy the equation. Here's a detailed breakdown of each step:
1. Eliminate the Fractions
The first step in solving the equation 4/(w-1) - 3/(w+2) = 5/w is to eliminate the fractions. We achieve this by multiplying both sides of the equation by the least common denominator (LCD), which, as we identified earlier, is w(w-1)(w+2). This multiplication will clear the denominators, transforming the equation into a more manageable polynomial equation. When we multiply each term by the LCD, we are essentially multiplying each fraction by the factors that are missing from its own denominator. This process cancels out the denominators, leaving us with a linear combination of polynomials.
So, let's perform the multiplication:
w(w-1)(w+2) * [4/(w-1) - 3/(w+2)] = w(w-1)(w+2) * [5/w]
This distributes to:
4w(w+2) - 3w(w-1) = 5(w-1)(w+2)
This step is crucial as it transforms a complex rational equation into a simpler polynomial equation, making it easier to solve. By carefully multiplying each term by the LCD, we ensure that the equation remains balanced and that we are one step closer to finding the solution.
2. Expand and Simplify
Following the elimination of fractions, the next crucial step involves expanding and simplifying the equation. This process entails distributing the terms and combining like terms to obtain a simplified polynomial equation. Starting from the equation obtained in the previous step:
4w(w+2) - 3w(w-1) = 5(w-1)(w+2)
We first expand each term:
4w^2 + 8w - 3w^2 + 3w = 5(w^2 + 2w - w - 2)
Now, simplify both sides:
w^2 + 11w = 5(w^2 + w - 2)
Continue expanding on the right side:
w^2 + 11w = 5w^2 + 5w - 10
This step is vital in condensing the equation into its simplest form, making it easier to identify the type of equation we are dealing with and the appropriate method for solving it. By carefully expanding and simplifying, we pave the way for the subsequent steps in finding the solution.
3. Rearrange into Quadratic Form
After expanding and simplifying, our next objective is to rearrange the equation into a standard quadratic form. This form is expressed as ax² + bx + c = 0, where a, b, and c are constants. Rearranging the equation into this form allows us to apply standard methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. From our previous step, we have:
w^2 + 11w = 5w^2 + 5w - 10
To rearrange, we want to set one side of the equation to zero. Let's subtract w² + 11w from both sides:
0 = 5w^2 + 5w - 10 - w^2 - 11w
Combine like terms:
0 = 4w^2 - 6w - 10
Now, we have a quadratic equation in the standard form. To simplify further, we can divide the entire equation by the greatest common divisor of the coefficients, which is 2:
0 = 2w^2 - 3w - 5
This simplified quadratic equation is now ready for solving. Rearranging into quadratic form is a critical step as it sets the stage for applying well-established techniques to find the roots of the equation. This methodical approach ensures accuracy and efficiency in solving the problem.
4. Solve the Quadratic Equation
With the equation now in the standard quadratic form, 2w² - 3w - 5 = 0, we can proceed to solve for w. There are several methods to solve a quadratic equation, including factoring, completing the square, and using the quadratic formula. In this case, factoring appears to be the most straightforward method. We are looking for two numbers that multiply to (2 * -5 = -10) and add up to -3. These numbers are -5 and 2. Thus, we can rewrite the middle term as follows:
2w^2 - 5w + 2w - 5 = 0
Now, we factor by grouping:
w(2w - 5) + 1(2w - 5) = 0
Factor out the common term (2w - 5):
(w + 1)(2w - 5) = 0
Setting each factor equal to zero gives us the possible solutions for w:
w + 1 = 0 or 2w - 5 = 0
Solving these linear equations:
w = -1 or w = 5/2
Therefore, the solutions to the quadratic equation are w = -1 and w = 5/2. This step is the heart of the solution process, where we find the values of the variable that satisfy the equation. By using factoring, we efficiently arrived at the solutions, which are now ready for verification.
5. Verify the Solutions
After finding the potential solutions for w, the final crucial step is to verify these solutions in the original equation. This step is essential to ensure that the solutions are valid and do not lead to any undefined expressions, such as division by zero. Our potential solutions are w = -1 and w = 5/2. The original equation is:
4/(w-1) - 3/(w+2) = 5/w
Let's first check w = -1:
4/(-1-1) - 3/(-1+2) = 5/(-1)
4/(-2) - 3/(1) = -5
-2 - 3 = -5
-5 = -5
The equation holds true for w = -1. Now, let's check w = 5/2:
4/(5/2-1) - 3/(5/2+2) = 5/(5/2)
4/(3/2) - 3/(9/2) = 2
8/3 - 6/9 = 2
8/3 - 2/3 = 2
6/3 = 2
2 = 2
The equation also holds true for w = 5/2. Since both solutions satisfy the original equation and do not violate any restrictions (i.e., they don't make any denominators zero), they are both valid solutions. Verification is a critical step in the problem-solving process, as it confirms the accuracy of our solutions and ensures that we have addressed the equation correctly.
Final Answer
In conclusion, by systematically eliminating fractions, simplifying the equation, rearranging it into quadratic form, solving for the variable, and verifying the solutions, we have successfully solved the equation 4/(w-1) - 3/(w+2) = 5/w. The solutions we found are w = -1 and w = 5/2. Both of these solutions satisfy the original equation and do not result in any undefined expressions. Therefore, our final answer is:
- w = -1
- w = 5/2
These values represent the complete solution set for the given equation. This step-by-step approach not only helps in solving the equation but also provides a clear understanding of the underlying mathematical principles and techniques involved. Mastering these techniques is essential for tackling more complex algebraic problems in the future. Remember, accuracy and verification are key to successful problem-solving in mathematics.
Conclusion
In this comprehensive guide, we've meticulously walked through the process of solving the equation 4/(w-1) - 3/(w+2) = 5/w. We've highlighted the importance of each step, from eliminating fractions to verifying the solutions, ensuring a clear and thorough understanding of the methods involved. The key takeaways from this exercise include the significance of finding the least common denominator, simplifying polynomial expressions, rearranging equations into standard forms, and the critical role of verification in ensuring the accuracy of the solutions. By mastering these techniques, you'll be well-equipped to tackle a wide range of algebraic equations with confidence.
This problem demonstrates the interconnectedness of various algebraic concepts and the importance of a systematic approach to problem-solving. Mathematics is not just about finding the right answer but also about understanding the process and reasoning behind it. As you continue your mathematical journey, remember to apply these principles and techniques to new challenges, and always strive for clarity, accuracy, and a deep understanding of the underlying concepts. With practice and perseverance, you'll find that solving complex equations becomes a rewarding and intellectually stimulating endeavor. We hope this guide has been helpful, and we encourage you to continue exploring the fascinating world of mathematics.
