Solving $\frac{4}{x}=\frac{1}{4 X}+3$ A Step-by-Step Guide

This article delves into the process of solving the equation $\frac{4}{x} = \frac{1}{4x} + 3$. We will break down each step in detail, providing a clear and concise explanation to ensure a thorough understanding of the solution. This comprehensive guide aims to equip you with the necessary skills to tackle similar algebraic problems with confidence. From identifying the domain restrictions to verifying the final solution, we'll cover all the essential aspects of solving this equation. So, let's embark on this mathematical journey and unravel the intricacies of this problem.

Understanding the Equation

At the heart of our discussion is the equation $\frac{4}{x} = \frac{1}{4x} + 3$. This is a rational equation, which means it involves fractions where the variable, in this case x, appears in the denominator. Solving rational equations requires careful consideration of potential restrictions on the variable's values. Our primary goal is to find the value(s) of x that satisfy this equation, making the left-hand side equal to the right-hand side. However, we must first address the crucial aspect of domain restrictions. In this context, a domain restriction arises when a value of x would make the denominator of any fraction in the equation equal to zero, which is mathematically undefined. Examining our equation, we observe that x appears in the denominators of both fractions, $\frac{4}{x}$ and $\frac{1}{4x}$. Therefore, x cannot be zero. This is because substituting x = 0 into either fraction would result in division by zero, an operation that is not permitted in mathematics. We denote this restriction as x ≠ 0. This restriction is paramount because any solution we find must adhere to this condition. If we arrive at a solution that contradicts this restriction, we must discard it as an extraneous solution. Ignoring domain restrictions can lead to incorrect answers and a misunderstanding of the equation's behavior. Therefore, it's vital to identify these restrictions at the outset and keep them in mind throughout the solving process. With the domain restriction firmly established, we can proceed to manipulate the equation algebraically to isolate x and determine its value(s). Remember, the key is to maintain the equality throughout the steps, ensuring that any operation performed on one side of the equation is also applied to the other side. This balance is essential to arrive at a valid solution.

Eliminating the Fractions

To effectively solve the equation \frac{4}{x} = rac{1}{4x} + 3, the initial step involves eliminating the fractions. This simplification makes the equation easier to manipulate and solve. The most common technique for achieving this is to multiply both sides of the equation by the least common denominator (LCD) of all the fractions present. In our equation, the denominators are x and 4x. The LCD is the smallest expression that is divisible by both x and 4x. In this case, the LCD is 4x. Multiplying both sides of the equation by 4x will clear the fractions. This process is based on the fundamental property of equality: if you multiply both sides of an equation by the same non-zero value, the equality remains valid. This is a critical principle in algebraic manipulations. Let's perform the multiplication: 4x * ($\frac{4}{x}$) = 4x * ($\frac{1}{4x}$ + 3). On the left-hand side, the x in the numerator and denominator cancel out, leaving us with 4 * 4 = 16. On the right-hand side, we need to distribute the 4x to both terms inside the parentheses. This gives us 4x * ($\frac{1}{4x}$) + 4x * 3. In the first term, the 4x in the numerator and denominator cancel out, leaving us with 1. In the second term, we have 4x * 3 = 12x. So, after multiplying both sides by the LCD and simplifying, our equation transforms into 16 = 1 + 12x. This is a significant step because we've converted the original rational equation into a simpler linear equation, which is much easier to solve. By eliminating the fractions, we've cleared the way for isolating the variable x and finding its value. The next step will involve rearranging the terms in the equation to bring the x term to one side and the constant terms to the other. This is a standard procedure in solving linear equations and will lead us closer to the final solution.

Isolating the Variable

With the equation simplified to 16 = 1 + 12x, the next crucial step is to isolate the variable x. This involves rearranging the terms in the equation so that the term containing x is on one side of the equation and the constant terms are on the other side. The fundamental principle we rely on here is the addition property of equality: you can add or subtract the same value from both sides of an equation without changing the equality. To isolate the term with x, which is 12x, we need to eliminate the constant term, 1, from the right-hand side of the equation. We can achieve this by subtracting 1 from both sides of the equation. This gives us 16 - 1 = 1 + 12x - 1. Simplifying both sides, we get 15 = 12x. Now, the equation has been transformed to a form where the term with x is isolated on the right-hand side, and a constant term is on the left-hand side. The next step is to isolate x itself. Currently, x is being multiplied by 12. To undo this multiplication and isolate x, we need to perform the inverse operation, which is division. We will divide both sides of the equation by 12. This is based on the division property of equality: you can divide both sides of an equation by the same non-zero value without changing the equality. Dividing both sides by 12, we get $\frac{15}{12}$ = $\frac{12x}{12}$. Simplifying, we have x = $\frac{15}{12}$. The fraction $\frac{15}{12}$ can be further simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This gives us x = $\frac{5}{4}$. At this point, we have found a potential solution for x. However, it's crucial to remember the domain restriction we identified at the beginning: x ≠ 0. Our solution, x = $\frac{5}{4}$, does not violate this restriction, so it remains a valid candidate. The final step is to verify this solution by substituting it back into the original equation to ensure that it satisfies the equation.

Verifying the Solution

Having obtained a potential solution, x = $\frac{5}{4}$, it is crucial to verify its validity by substituting it back into the original equation, $\frac{4}{x} = \frac{1}{4x} + 3$. This step is essential to confirm that our solution satisfies the equation and is not an extraneous solution, which can arise during the process of solving rational equations. Substituting x = $\frac{5}{4}$ into the left-hand side (LHS) of the equation, we get LHS = \frac{4}{rac{5}{4}}. To divide by a fraction, we multiply by its reciprocal, so LHS = 4 * $\frac{4}{5}$ = $\frac{16}{5}$. Now, let's substitute x = $\frac{5}{4}$ into the right-hand side (RHS) of the equation. We get RHS = \frac{1}{4 * rac{5}{4}} + 3. Simplifying the denominator, we have 4 * $\frac{5}{4}$ = 5, so RHS = $\frac{1}{5}$ + 3. To add these terms, we need a common denominator, which is 5. So, we rewrite 3 as $\frac{15}{5}$, and we have RHS = $\frac{1}{5}$ + $\frac{15}{5}$ = $\frac{16}{5}$. Comparing the LHS and RHS, we see that LHS = $\frac{16}{5}$ and RHS = $\frac{16}{5}$. Since LHS = RHS, our solution x = $\frac{5}{4}$ satisfies the original equation. This confirms that x = $\frac{5}{4}$ is indeed a valid solution. The verification step is a vital part of the problem-solving process, especially when dealing with rational equations. It ensures that the solution we have obtained is correct and that no errors were introduced during the algebraic manipulations. By substituting the solution back into the original equation, we can have confidence in our answer. In summary, we have successfully solved the equation $\frac{4}{x} = \frac{1}{4x} + 3$ and verified that the solution is x = $\frac{5}{4}$.

Conclusion

In this comprehensive guide, we meticulously solved the equation $\frac{4}{x} = \frac{1}{4x} + 3$. We began by understanding the equation and identifying the crucial domain restriction, x ≠ 0. This restriction stemmed from the presence of x in the denominators of the fractions, which prohibits x from being zero to avoid division by zero. Next, we eliminated the fractions by multiplying both sides of the equation by the least common denominator, 4x, which transformed the equation into a simpler linear form: 16 = 1 + 12x. We then proceeded to isolate the variable x by subtracting 1 from both sides and subsequently dividing both sides by 12, leading us to the potential solution x = $\frac{5}{4}$. To ensure the validity of our solution, we meticulously verified it by substituting x = $\frac{5}{4}$ back into the original equation. This verification process confirmed that our solution satisfied the equation, demonstrating that x = $\frac{5}{4}$ is indeed a valid solution. Throughout this process, we emphasized the importance of understanding domain restrictions and the need for verification, particularly when dealing with rational equations. These steps are crucial for avoiding extraneous solutions and ensuring the accuracy of our results. The techniques and principles discussed in this guide can be applied to a wide range of algebraic problems, providing a solid foundation for further mathematical explorations. Solving equations is a fundamental skill in mathematics, and by mastering these techniques, you can confidently tackle more complex problems. Remember, the key to success in mathematics is a combination of understanding the concepts, practicing the techniques, and carefully verifying your solutions.