Solving For Z In N = Cz / (c + Z) A Step-by-Step Guide

Jul 8, 2025 by Admin 55 views

Solving for z in the Equation N = cz / (c + z) : A Comprehensive Guide

Introduction

In the realm of mathematics, manipulating equations to isolate specific variables is a fundamental skill. This article delves into the process of solving the equation N = cz / (c + z) for the variable z. This type of equation often appears in various fields, including physics, engineering, and economics, where understanding the relationship between variables is crucial. Our goal is to provide a step-by-step guide that not only solves for z but also elucidates the underlying principles of algebraic manipulation. By the end of this article, you will have a solid understanding of how to tackle similar equations and extract the desired variable.

Understanding the Equation

Before diving into the solution, let's first dissect the equation N = cz / (c + z). This equation expresses N in terms of c and z. Here, N represents the dependent variable, while c and z are independent variables. The equation involves a fraction, which means our initial steps will focus on eliminating the denominator to simplify the expression. The presence of z in both the numerator and the denominator adds a layer of complexity, requiring careful algebraic manipulation. Understanding the structure of the equation is paramount to choosing the right approach for isolating z. We need to consider the order of operations and how to reverse them effectively. The key is to strategically apply algebraic operations that maintain the equality while gradually bringing z to one side of the equation.

Step-by-Step Solution

Step 1: Eliminate the Denominator

The first step in solving for z is to eliminate the denominator. To do this, we multiply both sides of the equation by (c + z):

N(c + z) = cz

This step is crucial because it removes the fraction, making the equation easier to manipulate. By multiplying both sides by the same expression, we maintain the balance of the equation, a fundamental principle in algebra. This operation sets the stage for the next steps, where we will distribute and rearrange terms to isolate z. Remember, the goal is to get all terms containing z on one side and all other terms on the other side.

Step 2: Distribute on the Left Side

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

Nc + Nz = cz

This step expands the left side, revealing the individual terms and preparing the equation for further manipulation. Distribution is a key algebraic technique that allows us to break down complex expressions into simpler components. Now, we have a linear equation in terms of z, which means we can proceed with collecting terms and isolating z. The distributive property is a cornerstone of algebraic manipulation, and mastering it is essential for solving a wide range of equations.

Step 3: Rearrange Terms to Isolate z

Now, we want to get all terms containing z on one side of the equation. Subtract Nz from both sides:

Nc = cz - Nz

This step is a crucial rearrangement that brings us closer to isolating z. By subtracting Nz from both sides, we maintain the equality while grouping the z terms together. This maneuver allows us to factor out z in the next step, a critical technique for solving for a variable that appears in multiple terms. The ability to strategically rearrange terms is a hallmark of algebraic proficiency.

Step 4: Factor out z

Factor out z from the right side:

Nc = z(c - N)

Factoring is the reverse process of distribution and is a powerful tool for isolating variables. By factoring out z, we transform the right side into a product of z and (c - N). This step is essential because it allows us to isolate z by dividing both sides by (c - N) in the next step. Factoring is a fundamental algebraic skill that is used extensively in solving equations and simplifying expressions.

Step 5: Solve for z

Finally, divide both sides by (c - N) to solve for z:

z = Nc / (c - N)

This is the solution for z in terms of N and c. We have successfully isolated z by performing a series of algebraic manipulations. This final step reveals the explicit relationship between z, N, and c. It's important to note that this solution is valid as long as c ≠ N, as division by zero is undefined. Understanding these limitations is crucial for applying the solution in real-world contexts.

Verification of the Solution

To ensure the correctness of our solution, we can substitute z = Nc / (c - N) back into the original equation and verify that it holds true. This process involves substituting the expression for z into the original equation and simplifying to see if both sides are equal. Verification is a crucial step in problem-solving, as it provides confidence in the accuracy of the solution and helps identify any potential errors. By performing this check, we can be certain that our solution is correct and applicable.

Substituting the Solution

Substitute z = Nc / (c - N) into the original equation N = cz / (c + z):

N = c[Nc / (c - N)] / {c + [Nc / (c - N)]}

This substitution replaces z with its equivalent expression in terms of N and c. The next step is to simplify this complex fraction to see if it reduces to an identity. Simplification involves performing algebraic operations such as combining fractions and canceling common factors. The goal is to show that the left side of the equation equals the right side, thereby confirming the validity of our solution.

Simplifying the Expression

To simplify, first, find a common denominator for the denominator:

N = (c²N / (c - N)) / [(c(c - N) + Nc) / (c - N)]

This step involves finding a common denominator for the terms in the denominator of the main fraction. This allows us to combine the terms and simplify the expression. The common denominator in this case is (c - N). This step is a crucial technique in simplifying complex fractions and is a common practice in algebraic manipulation.

Further simplification gives:

N = (c²N / (c - N)) / (c² / (c - N))

Here, we have simplified the denominator by combining like terms. The expression now has a simpler form, which makes it easier to proceed with the simplification. This step demonstrates the power of algebraic manipulation in reducing complex expressions to more manageable forms.

Final Simplification

Now, divide the fractions:

N = (c²N / (c - N)) * ((c - N) / c²)

Dividing by a fraction is equivalent to multiplying by its reciprocal. This step transforms the division problem into a multiplication problem, which is often easier to handle. This is a fundamental rule of fraction manipulation and is essential for simplifying complex expressions.

Cancel out common terms:

N = N

This final simplification shows that the left side of the equation is equal to the right side, confirming the correctness of our solution. The cancellation of common factors demonstrates the elegance of algebraic manipulation in arriving at a simple identity. This verification step provides confidence in the accuracy of the solution and ensures that it is valid.

Conclusion

In this comprehensive guide, we have successfully solved the equation N = cz / (c + z) for z, arriving at the solution z = Nc / (c - N). We have also verified the solution by substituting it back into the original equation and confirming its validity. This exercise highlights the importance of algebraic manipulation skills in solving equations and understanding the relationships between variables. The step-by-step approach we have taken can be applied to a wide range of similar problems. Mastering these techniques is crucial for success in mathematics and related fields.

Key Takeaways

Eliminating the Denominator: Multiplying both sides by the denominator is often the first step in solving equations involving fractions.
Distribution and Factoring: These are fundamental algebraic techniques for expanding and simplifying expressions.
Rearranging Terms: Strategically moving terms around the equation is crucial for isolating the desired variable.
Verification: Always verify your solution by substituting it back into the original equation.

By understanding and applying these key takeaways, you can confidently tackle a variety of algebraic problems and deepen your understanding of mathematical principles. The ability to solve equations is a valuable skill that is applicable in many areas of life and work.

This article provides a thorough guide on solving for z in the given equation, emphasizing the importance of each step and the underlying principles of algebraic manipulation. By following this guide, readers can gain a deeper understanding of how to solve similar equations and apply these skills in various contexts.