Solving The Equation 4/(5b) + 5/b = 7 A Step-by-Step Guide

In this article, we will delve into the step-by-step process of solving the equation $\frac{4}{5b} + \frac{5}{b} = 7$. This type of equation, involving fractions with variables in the denominator, is commonly encountered in algebra. Understanding how to solve it is crucial for mastering algebraic manipulations and problem-solving. We will break down the solution process into manageable steps, ensuring clarity and comprehension. This article is designed to be a comprehensive guide, suitable for students and anyone looking to refresh their algebra skills. By the end of this guide, you will be equipped with the knowledge and confidence to tackle similar equations effectively. We will also explore the underlying principles and concepts that make this solution possible, providing a deeper understanding of algebraic techniques. So, let's embark on this mathematical journey together and unravel the solution to this intriguing equation.

Understanding the Equation

Before diving into the solution, it's essential to understand the structure of the equation $\frac{4}{5b} + \frac{5}{b} = 7$. This equation is a rational equation, which means it involves fractions where the variable, b, appears in the denominator. Dealing with rational equations requires careful attention to avoid division by zero and to correctly manipulate the fractions. The key to solving this equation lies in combining the fractions on the left-hand side and then isolating the variable b. To achieve this, we need to find a common denominator for the fractions. The presence of b in the denominators of both fractions means that any value of b that makes the denominator zero is not a valid solution. Specifically, b cannot be equal to 0. This is an important consideration that we must keep in mind throughout the solution process. By understanding these fundamental aspects, we can approach the equation with a clear strategy and avoid potential pitfalls. We will now move on to the first step in solving the equation: finding the common denominator.

Finding the Common Denominator

The first crucial step in solving the equation $\frac{4}{5b} + \frac{5}{b} = 7$ is to identify the common denominator for the fractions on the left-hand side. The denominators are 5b and b. To find the least common denominator (LCD), we need to determine the smallest expression that is a multiple of both denominators. In this case, the LCD is 5b. To combine the fractions, we need to express each fraction with the common denominator. The first fraction, $\frac{4}{5b}$, already has the desired denominator. However, the second fraction, $\frac{5}{b}$, needs to be adjusted. To make the denominator of the second fraction 5b, we multiply both the numerator and the denominator by 5. This gives us $\frac{5 * 5}{5 * b} = \frac{25}{5b}$. Now that both fractions have the same denominator, we can combine them. This process of finding the common denominator is a fundamental technique in algebra, and it's essential for solving equations involving fractions. Once we have a common denominator, we can add the numerators and simplify the equation, bringing us closer to isolating the variable b. So, with the common denominator in hand, we are ready to proceed to the next step: combining the fractions.

Combining the Fractions

With the common denominator of 5b established, we can now combine the fractions in the equation $\frac{4}{5b} + \frac{5}{b} = 7$. We rewrote the equation as $\frac{4}{5b} + \frac{25}{5b} = 7$. Since both fractions now have the same denominator, we can add their numerators: $\frac{4 + 25}{5b} = 7$. This simplifies to $\frac{29}{5b} = 7$. Combining fractions is a crucial step in solving rational equations because it allows us to consolidate the terms on one side of the equation. By having a single fraction, we can then apply other algebraic techniques to isolate the variable. This step reduces the complexity of the equation and makes it easier to manipulate. Once the fractions are combined, we have a simpler equation to work with, which brings us closer to finding the value of b. The resulting equation, $\frac{29}{5b} = 7$, is now in a form that we can easily solve by multiplying both sides by 5b to eliminate the fraction. Thus, combining the fractions is a pivotal step in our journey to solve the equation.

Eliminating the Fraction

Now that we have the equation in the form $\frac{29}{5b} = 7$, the next step is to eliminate the fraction. To do this, we multiply both sides of the equation by the denominator, 5b. This gives us: $5b * \frac{29}{5b} = 7 * 5b$. On the left side, the 5b in the numerator and denominator cancel each other out, leaving us with 29. On the right side, we have $7 * 5b = 35b$. So, the equation becomes $29 = 35b$. Eliminating the fraction is a critical step because it transforms the rational equation into a simpler linear equation, which is much easier to solve. This technique is widely used in algebra to handle equations involving fractions. By multiplying both sides by the denominator, we effectively clear the fraction and set the stage for isolating the variable. This step simplifies the equation and brings us closer to the final solution. With the fraction eliminated, we now have a straightforward equation that we can solve for b. The next step involves isolating b to find its value.

Isolating the Variable

After eliminating the fraction, we have the equation $29 = 35b$. To isolate the variable b, we need to get b by itself on one side of the equation. This involves performing the inverse operation of what is being done to b. In this case, b is being multiplied by 35. The inverse operation of multiplication is division. Therefore, we divide both sides of the equation by 35: $\frac{29}{35} = \frac{35b}{35}$. On the right side, the 35s cancel out, leaving us with b. So, the equation becomes $b = \frac{29}{35}$. Isolating the variable is a fundamental concept in algebra, and it's the key to solving equations. By performing the same operation on both sides of the equation, we maintain the equality while moving closer to finding the value of the variable. This step brings us to the solution, which is the value of b that satisfies the original equation. Once we have isolated the variable, we have found the solution to the equation. The final step is to check the solution to ensure it is valid.

Checking the Solution

Now that we have found a potential solution, $b = \frac{29}{35}$, it is crucial to check whether this value satisfies the original equation, $\frac{4}{5b} + \frac{5}{b} = 7$. Substituting $b = \frac{29}{35}$ into the equation, we get: $\frac{4}{5(\frac{29}{35})} + \frac{5}{\frac{29}{35}} = 7$. First, let's simplify the first term: $\frac{4}{5(\frac{29}{35})} = \frac{4}{\frac{29}{7}} = \frac{4 * 7}{29} = \frac{28}{29}$. Next, let's simplify the second term: $\frac{5}{\frac{29}{35}} = \frac{5 * 35}{29} = \frac{175}{29}$. Now, we add the two fractions: $\frac{28}{29} + \frac{175}{29} = \frac{28 + 175}{29} = \frac{203}{29}$. Finally, we check if this equals 7: $\frac{203}{29} = 7$. Since 203 divided by 29 is indeed 7, our solution is correct. Checking the solution is an essential step in the problem-solving process. It ensures that we haven't made any errors in our calculations and that the solution we found is valid. By substituting the solution back into the original equation, we can verify its correctness and gain confidence in our answer. This step confirms that our solution is accurate and that we have successfully solved the equation.

Solution Set

After successfully solving the equation $\frac{4}{5b} + \frac{5}{b} = 7$ and verifying the solution, we can now express the solution set. The solution we found is $b = \frac{29}{35}$. The solution set is the set of all values that satisfy the equation. In this case, there is only one value that satisfies the equation. Therefore, the solution set is {$\frac{29}{35}$}. The solution set provides a concise way to present the answer to the problem. It is a standard notation in mathematics to express the solution(s) to an equation as a set. This notation clearly indicates the values of the variable that make the equation true. In this instance, the solution set contains a single element, which is the value of b that we calculated. This concludes our journey of solving the equation. We have walked through each step, from understanding the equation to verifying the solution, ensuring clarity and comprehension. The solution set is the final answer, representing the value of b that satisfies the given equation.

In conclusion, we have successfully solved the equation $\frac{4}{5b} + \frac{5}{b} = 7$ by following a series of logical steps. We began by understanding the equation and identifying the need for a common denominator. We then found the common denominator, combined the fractions, and eliminated the fraction by multiplying both sides by the denominator. This transformed the equation into a simpler form, allowing us to isolate the variable b. We divided both sides by 35 to isolate b, resulting in the solution $b = \frac{29}{35}$. Finally, we verified the solution by substituting it back into the original equation, confirming its validity. The solution set is {$\frac{29}{35}$}. This process demonstrates the importance of understanding algebraic principles and applying them systematically. Solving rational equations requires careful attention to detail and a solid grasp of algebraic techniques. By mastering these techniques, you can confidently tackle a wide range of algebraic problems. This comprehensive guide has provided a clear and step-by-step approach to solving this particular equation, and the principles discussed can be applied to solve other similar equations. Remember, practice is key to mastering algebra, so continue to challenge yourself with new problems and reinforce your understanding.