Solving The Equation (3 / (2b - 5)) - (4 / (b - 3)) = 0 A Step-by-Step Guide

Jul 11, 2025 by Jeany 77 views

Solving the Equation: A Comprehensive Guide to Finding the Solution of (3 / (2b - 5)) - (4 / (b - 3)) = 0

Introduction: Unveiling the intricacies of solving rational equations

In the realm of mathematics, solving equations is a fundamental skill. This article will delve into the process of solving the rational equation (3 / (2b - 5)) - (4 / (b - 3)) = 0. Rational equations, which involve fractions with variables in the denominator, often require careful manipulation to arrive at a solution. We will walk through the steps systematically, ensuring clarity and understanding along the way. Our main focus is to help you understand how to solve the equation effectively and efficiently.

1. Clearing the Fractions: The first step to simplification

The initial hurdle in solving this equation lies in the presence of fractions. To eliminate these fractions and simplify the equation, we need to find the least common denominator (LCD) of the denominators (2b - 5) and (b - 3). In this case, the LCD is simply the product of the two denominators, which is (2b - 5)(b - 3). Multiplying both sides of the equation by the LCD will clear the fractions. This process is crucial for transforming the equation into a more manageable form. We are essentially aiming to get rid of the fractions to work with a simpler algebraic expression. The importance of finding the correct LCD cannot be overstated, as it forms the basis for the subsequent steps in solving the equation.

2. Multiplying by the LCD: Eliminating denominators for a cleaner equation

Now, let's multiply both sides of the equation (3 / (2b - 5)) - (4 / (b - 3)) = 0 by the LCD, which is (2b - 5)(b - 3). This step is critical in removing the denominators and paving the way for further simplification. When we multiply the first term, (3 / (2b - 5)), by the LCD, the (2b - 5) terms cancel out, leaving us with 3(b - 3). Similarly, when we multiply the second term, (4 / (b - 3)), by the LCD, the (b - 3) terms cancel out, leaving us with -4(2b - 5). On the right side of the equation, multiplying 0 by the LCD simply results in 0. This entire process is a standard technique in solving the equation and dealing with rational expressions. The resulting equation is free of fractions, making it easier to handle.

3. Expanding and Simplifying: Reducing the equation to its simplest form

After multiplying by the LCD, our equation now looks like 3(b - 3) - 4(2b - 5) = 0. The next step is to expand the expressions on the left side of the equation. Distribute the 3 across (b - 3) to get 3b - 9. Similarly, distribute the -4 across (2b - 5) to get -8b + 20. The equation now becomes 3b - 9 - 8b + 20 = 0. Now, we need to combine like terms. Combine the 'b' terms (3b and -8b) to get -5b. Combine the constant terms (-9 and +20) to get +11. The equation is now simplified to -5b + 11 = 0. This simplification process is crucial for solving the equation as it reduces the complexity and brings us closer to isolating the variable 'b'.

4. Isolating the Variable: The key to finding the solution

Now that we have the simplified equation -5b + 11 = 0, our goal is to isolate the variable 'b'. To do this, we first need to get the term with 'b' by itself on one side of the equation. Subtract 11 from both sides of the equation. This gives us -5b = -11. Next, to completely isolate 'b', we need to divide both sides of the equation by -5. This gives us b = -11 / -5, which simplifies to b = 11/5. The process of isolating the variable is a cornerstone of solving the equation, and it involves using inverse operations to undo the operations that are being applied to the variable.

5. The Solution: Presenting the value of b

After performing the algebraic manipulations, we have arrived at the solution for the equation. The value of 'b' that satisfies the equation (3 / (2b - 5)) - (4 / (b - 3)) = 0 is b = 11/5. This means that if we substitute 11/5 for 'b' in the original equation, both sides of the equation will be equal. It's important to clearly state the solution, as it is the culmination of the solving the equation process. We have successfully found the value of the unknown variable.

6. Checking the Solution: Ensuring accuracy and validity

Before definitively declaring our solution, it's crucial to check whether it is valid. In the context of rational equations, we need to ensure that our solution does not make any of the denominators in the original equation equal to zero. If a solution makes a denominator zero, it is an extraneous solution and must be discarded. Let's substitute b = 11/5 into the denominators (2b - 5) and (b - 3). For (2b - 5), we have 2(11/5) - 5 = 22/5 - 25/5 = -3/5, which is not zero. For (b - 3), we have 11/5 - 3 = 11/5 - 15/5 = -4/5, which is also not zero. Since our solution does not make any denominator zero, it is a valid solution. This step of checking the solution is essential in solving the equation and ensuring the accuracy of our result.

7. Verification: Substituting the solution back into the original equation

To be absolutely certain of our solution, we can perform a final verification step. Substitute b = 11/5 back into the original equation (3 / (2b - 5)) - (4 / (b - 3)) = 0 and check if both sides of the equation are equal. Substituting b = 11/5, we get (3 / (2(11/5) - 5)) - (4 / (11/5 - 3)). Simplifying the denominators, we get (3 / (-3/5)) - (4 / (-4/5)). Dividing by a fraction is the same as multiplying by its reciprocal, so we have 3 * (-5/3) - 4 * (-5/4). This simplifies to -5 - (-5) = -5 + 5 = 0, which equals the right side of the original equation. This verification step provides further confirmation that our solution is correct. It's a valuable practice in solving the equation to ensure the reliability of the answer.

Conclusion: Mastering the art of solving rational equations

In conclusion, we have successfully solved the equation (3 / (2b - 5)) - (4 / (b - 3)) = 0. The solution we found is b = 11/5. We followed a systematic approach, including clearing fractions, simplifying the equation, isolating the variable, and checking the solution. Solving the equation like this requires careful attention to detail and a solid understanding of algebraic principles. By mastering these techniques, you can confidently tackle a wide range of rational equations. This step-by-step guide has demonstrated the process thoroughly, ensuring that you have the knowledge and skills to solve similar equations in the future. Remember, practice is key to mastering mathematical concepts, so continue to apply these principles to various problems.