Solving The Equation (5(2n+1))/4 = 11 A Step-by-Step Guide

Understanding the Problem

In this mathematical exploration, we are presented with the equation ${\frac{5(2n+1)}{4} = 11}$ and our main goal is to isolate the variable 'n' to find its value. This involves a series of algebraic manipulations, each step carefully designed to maintain the equality of the equation while bringing us closer to the solution. To effectively solve this equation, we will delve into the fundamental principles of algebra, employing techniques such as distribution, multiplication, and subtraction. By meticulously applying these operations, we will systematically unravel the equation, ultimately determining the numerical value that satisfies the given condition. The process of solving equations like this is a cornerstone of mathematics, finding applications in numerous fields ranging from engineering to economics. Therefore, a firm understanding of the techniques involved is not only crucial for academic success but also for practical problem-solving in the real world. In the following sections, we will break down each step of the solution in detail, ensuring clarity and a thorough understanding of the underlying concepts. This approach will not only help in solving the specific equation at hand but also equip you with the skills to tackle similar algebraic challenges with confidence.

Step-by-Step Solution

The task at hand is to solve the equation ${\frac{5(2n+1)}{4} = 11}$, and to begin this algebraic journey, we must first eliminate the fraction. The most effective way to achieve this is by multiplying both sides of the equation by 4. This operation is rooted in the fundamental principle that if two quantities are equal, multiplying both by the same number will preserve that equality. By multiplying both sides by 4, we effectively cancel out the denominator on the left side, simplifying the equation and bringing us closer to isolating the variable 'n'. This is a crucial step because it transforms the equation from a complex fractional form into a more manageable linear form. Once the fraction is eliminated, we can proceed with further algebraic manipulations to isolate 'n'. This process of eliminating fractions is a common technique in solving equations, and it is essential to master it for success in algebra and beyond. The next phase will involve distributing the 5 on the left side and then further isolating 'n' by using inverse operations. As we move forward, each step will be carefully explained, ensuring that you understand the logic and rationale behind each manipulation. Let's proceed with the next step, where we distribute the 5 and continue our quest to find the value of 'n'.

Step 1: Multiply Both Sides by 4

To eliminate the fraction, we multiply both sides of the equation by 4:

$$4 imes \frac{5(2n+1)}{4} = 11 imes 4$$

This simplifies to:

$$5(2n+1) = 44$$

Step 2: Distribute the 5

Next, we distribute the 5 on the left side of the equation:

$$5 imes 2n + 5 imes 1 = 44$$

This results in:

$$10n + 5 = 44$$

Step 3: Subtract 5 from Both Sides

To isolate the term with 'n', we subtract 5 from both sides:

$$10n + 5 - 5 = 44 - 5$$

Which simplifies to:

$$10n = 39$$

Step 4: Divide Both Sides by 10

Finally, to solve for 'n', we divide both sides by 10:

$$\frac{10n}{10} = \frac{39}{10}$$

This gives us the solution:

$$n = \frac{39}{10}$$

Verification

Ensuring the accuracy of our solution is a critical aspect of the problem-solving process. In mathematics, simply arriving at an answer is not sufficient; we must also verify that the answer satisfies the original conditions of the problem. In the context of solving equations, this means substituting the value we have calculated back into the original equation and confirming that both sides of the equation are indeed equal. This process serves as a safeguard against errors that may have occurred during the algebraic manipulations. It provides us with a high degree of confidence that our solution is correct. In this instance, we will substitute ${n = \frac{39}{10}}$ back into the original equation ${\frac{5(2n+1)}{4} = 11}$. By performing the necessary arithmetic operations, we will evaluate both the left-hand side (LHS) and the right-hand side (RHS) of the equation. If the LHS equals the RHS, then our solution is verified. If not, we would need to revisit our steps to identify and correct any errors. This rigorous verification step is a hallmark of mathematical rigor and is essential for developing a deep understanding of the subject. It not only confirms the correctness of our answer but also reinforces our understanding of the underlying principles and techniques used in solving the equation.

To verify the solution, substitute ${n = \frac{39}{10}}$ back into the original equation:

$$\frac{5(2(\frac{39}{10})+1)}{4} = 11$$

Let's simplify the left side:

$$\frac{5(\frac{78}{10}+1)}{4} = \frac{5(\frac{78}{10}+\frac{10}{10})}{4}$$

$$\frac{5(\frac{88}{10})}{4} = \frac{5 \times \frac{88}{10}}{4}$$

$$\frac{\frac{440}{10}}{4} = \frac{44}{4}$$

$$11 = 11$$

The solution is verified.

Conclusion

In summary, we embarked on a mathematical journey to solve the equation ${\frac{5(2n+1)}{4} = 11}$. Through a series of carefully executed algebraic steps, we successfully isolated the variable 'n' and determined its value. The process began with eliminating the fraction by multiplying both sides of the equation by 4, a crucial step that simplified the equation and paved the way for further manipulations. Following this, we distributed the 5 on the left side, subtracted 5 from both sides to isolate the term containing 'n', and finally, divided both sides by 10 to solve for 'n'. This systematic approach, rooted in the fundamental principles of algebra, allowed us to arrive at the solution ${n = \frac{39}{10}}$. However, our task was not complete with just finding the solution. We recognized the paramount importance of verification in mathematics. To ensure the accuracy of our result, we substituted the calculated value of 'n' back into the original equation. This step served as a rigorous check, confirming that our solution satisfied the initial conditions of the problem. The verification process not only instilled confidence in our answer but also reinforced our understanding of the algebraic manipulations we had performed. This comprehensive approach, encompassing both the solution and its verification, is a hallmark of mathematical rigor and is essential for developing a deep and lasting understanding of the subject. By mastering these techniques, we can confidently tackle a wide range of algebraic challenges, both in academic settings and in real-world applications. The journey through this problem has not only provided us with a solution but has also enhanced our problem-solving skills and our appreciation for the power and elegance of mathematics.

Therefore, the solution to the equation ${\frac{5(2n+1)}{4} = 11}$ is:

$$n = \frac{39}{10}$$