Subtract And Simplify The Expression \( \frac{1}{2z} - \frac{z}{2z-1} \). Which Option Is Correct? A. \( \frac{2z^2 - 2z - 1}{4z^2 + 2z} \) B. \( \frac{-2z^2 + 2z - 1}{4z^2 - 2z} \) C. \( \frac{-2z^2 + 2z + 1}{4z^2 + 2z} \) D. \( \frac{2z^2 - 2z + 1}{4z^2 - 2z} \)

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In the realm of mathematics, particularly in algebra, simplifying expressions is a fundamental skill. This article delves into the process of subtracting and simplifying the expression 12z−z2z−1{\frac{1}{2z} - \frac{z}{2z-1}}. We will break down each step, ensuring a clear understanding of the methodology involved. This detailed explanation is designed to help students, educators, and anyone interested in enhancing their algebraic skills. By the end of this guide, you will not only be able to solve this specific problem but also gain a broader understanding of how to tackle similar algebraic challenges. Let's embark on this mathematical journey together, unraveling the complexities and revealing the elegance of algebraic simplification.

Understanding the Problem

To effectively subtract and simplify the expression, it's crucial to first understand the problem's components. Our task is to subtract the fraction z2z−1{\frac{z}{2z-1}} from 12z{\frac{1}{2z}}. This involves finding a common denominator, performing the subtraction, and then simplifying the resulting expression, if possible. The given options provide potential solutions, and our goal is to arrive at the correct one through a step-by-step process. Before we dive into the solution, let’s discuss the foundational concepts that underpin this algebraic manipulation. This includes understanding fractions, algebraic expressions, and the rules governing their addition and subtraction. These principles are essential for navigating the complexities of this problem and similar algebraic challenges. Grasping these basics will not only help in solving this particular question but will also strengthen your overall mathematical foundation. So, let’s begin by reinforcing these key concepts to ensure a solid understanding before we proceed with the solution.

Finding a Common Denominator

The cornerstone of subtracting fractions lies in finding a common denominator. In our case, we need to find a common denominator for 2z{2z} and 2z−1{2z-1}. Since these two expressions do not share any common factors, the least common denominator (LCD) will be their product, which is 2z(2z−1){2z(2z-1)}. Once we have the common denominator, we need to rewrite each fraction with this new denominator. This involves multiplying the numerator and denominator of each fraction by the appropriate factor. For the first fraction, 12z{\frac{1}{2z}}, we multiply both the numerator and the denominator by (2z−1){(2z-1)}. For the second fraction, z2z−1{\frac{z}{2z-1}}, we multiply both the numerator and the denominator by 2z{2z}. This process ensures that we are working with equivalent fractions that can be easily subtracted. Understanding and mastering this step is crucial, as it forms the basis for many algebraic manipulations involving fractions. Let’s proceed with this step to transform our original expression into a form where subtraction can be directly applied.

Rewriting the Fractions

Having identified the common denominator as 2z(2z−1){2z(2z-1)}, our next step is to rewrite each fraction with this denominator. For the first fraction, 12z{\frac{1}{2z}}, we multiply both the numerator and the denominator by (2z−1){(2z-1)}, resulting in:

12z×2z−12z−1=2z−12z(2z−1){ \frac{1}{2z} \times \frac{2z-1}{2z-1} = \frac{2z-1}{2z(2z-1)} }

Similarly, for the second fraction, z2z−1{\frac{z}{2z-1}}, we multiply both the numerator and the denominator by 2z{2z}, which gives us:

z2z−1×2z2z=2z22z(2z−1){ \frac{z}{2z-1} \times \frac{2z}{2z} = \frac{2z^2}{2z(2z-1)} }

Now, both fractions have the same denominator, allowing us to perform the subtraction. This is a critical step in simplifying the expression, as it sets the stage for combining the numerators. The ability to manipulate fractions in this way is a fundamental skill in algebra, and it's essential for solving a wide range of mathematical problems. With the fractions rewritten, we are now ready to subtract them and move closer to our final simplified form. Let’s proceed with the subtraction in the next section.

Subtracting the Fractions

With both fractions now sharing a common denominator of 2z(2z−1){2z(2z-1)}, we can proceed with the subtraction. This involves subtracting the numerators while keeping the denominator the same. So, we have:

2z−12z(2z−1)−2z22z(2z−1)=(2z−1)−2z22z(2z−1){ \frac{2z-1}{2z(2z-1)} - \frac{2z^2}{2z(2z-1)} = \frac{(2z-1) - 2z^2}{2z(2z-1)} }

Now, we need to simplify the numerator by combining like terms. The numerator is (2z−1)−2z2{(2z-1) - 2z^2}, which can be rearranged as ext{−2z2+2z−1{-2z^2 + 2z - 1}}. This step is crucial for reducing the expression to its simplest form. The ability to accurately subtract fractions and simplify the resulting numerator is a key skill in algebra. It requires careful attention to detail and a solid understanding of algebraic operations. Now that we have subtracted the fractions and simplified the numerator, the next step is to expand the denominator and see if further simplification is possible. Let’s move on to expanding the denominator and exploring potential simplifications.

Expanding and Simplifying

After subtracting the fractions, our expression looks like this:

−2z2+2z−12z(2z−1){ \frac{-2z^2 + 2z - 1}{2z(2z-1)} }

Now, we need to expand the denominator, which is 2z(2z−1){2z(2z-1)}. Multiplying this out, we get:

2z(2z−1)=4z2−2z{ 2z(2z-1) = 4z^2 - 2z }

So, our expression now becomes:

−2z2+2z−14z2−2z{ \frac{-2z^2 + 2z - 1}{4z^2 - 2z} }

At this point, we should check if the expression can be further simplified. We look for common factors between the numerator and the denominator. In this case, there are no common factors that can be canceled out. Therefore, the expression is in its simplest form. This step is a critical part of the simplification process, as it ensures that the final answer is in its most concise form. The ability to recognize when an expression is fully simplified is a key skill in algebra. Having expanded the denominator and checked for further simplifications, we have arrived at our final simplified expression. Now, let's compare our result with the given options to identify the correct answer.

Comparing with the Options

Our simplified expression is:

−2z2+2z−14z2−2z{ \frac{-2z^2 + 2z - 1}{4z^2 - 2z} }

Now, let's compare this with the given options:

A. 2z2−2z−14z2+2z{\frac{2z^2 - 2z - 1}{4z^2 + 2z}}

B. −2z2+2z−14z2−2z{\frac{-2z^2 + 2z - 1}{4z^2 - 2z}}

C. −2z2+2z+14z2+2z{\frac{-2z^2 + 2z + 1}{4z^2 + 2z}}

D. 2z2−2z+14z2−2z{\frac{2z^2 - 2z + 1}{4z^2 - 2z}}

By comparing our simplified expression with the options, we can see that option B matches our result exactly. This step is crucial to ensure that we select the correct answer from the provided choices. It involves careful comparison and attention to detail to avoid any errors. The ability to accurately compare and match expressions is an essential skill in mathematics. We have now successfully matched our simplified expression with one of the options. In the next section, we will state the final answer and summarize the steps we took to arrive at the solution.

Final Answer

After meticulously working through the steps of finding a common denominator, subtracting the fractions, expanding, and simplifying, we arrived at the expression:

−2z2+2z−14z2−2z{ \frac{-2z^2 + 2z - 1}{4z^2 - 2z} }

Comparing this with the given options, we find that it matches option B.

Therefore, the final answer is:

B. −2z2+2z−14z2−2z{\frac{-2z^2 + 2z - 1}{4z^2 - 2z}}

This final step is the culmination of our efforts, where we confidently state the solution we have derived. It's a moment of validation, confirming the accuracy of our work. Now, let's summarize the entire process to reinforce our understanding and provide a clear recap of the steps involved.

Summary of Steps

To subtract and simplify the expression 12z−z2z−1{\frac{1}{2z} - \frac{z}{2z-1}}, we followed these steps:

  1. Found a common denominator: The common denominator for 2z{2z} and 2z−1{2z-1} is 2z(2z−1){2z(2z-1)}.

  2. Rewrote the fractions:

    • 12z{\frac{1}{2z}} became 2z−12z(2z−1){\frac{2z-1}{2z(2z-1)}}
    • z2z−1{\frac{z}{2z-1}} became 2z22z(2z−1){\frac{2z^2}{2z(2z-1)}}

  3. Subtracted the fractions:

    2z−12z(2z−1)−2z22z(2z−1)=(2z−1)−2z22z(2z−1){ \frac{2z-1}{2z(2z-1)} - \frac{2z^2}{2z(2z-1)} = \frac{(2z-1) - 2z^2}{2z(2z-1)} }

  4. Simplified the numerator: The numerator (2z−1)−2z2{(2z-1) - 2z^2} simplified to −2z2+2z−1{-2z^2 + 2z - 1}.

  5. Expanded the denominator: The denominator 2z(2z−1){2z(2z-1)} expanded to 4z2−2z{4z^2 - 2z}.

  6. Combined the results: The expression became −2z2+2z−14z2−2z{\frac{-2z^2 + 2z - 1}{4z^2 - 2z}}.

  7. Compared with options: We matched our result with option B.

This step-by-step summary provides a clear and concise overview of the entire solution process. It serves as a valuable tool for reinforcing understanding and can be used as a reference for tackling similar problems in the future. By breaking down the problem into manageable steps, we have demonstrated a systematic approach to solving algebraic expressions. This approach can be applied to a wide range of mathematical problems, making it a valuable skill to develop. With this summary, we conclude our comprehensive guide to subtracting and simplifying the given expression.

Conclusion

In conclusion, we have successfully subtracted and simplified the expression 12z−z2z−1{\frac{1}{2z} - \frac{z}{2z-1}}, arriving at the final answer B. −2z2+2z−14z2−2z{\frac{-2z^2 + 2z - 1}{4z^2 - 2z}}. This journey through algebraic manipulation has highlighted the importance of several key concepts, including finding a common denominator, rewriting fractions, subtracting numerators, expanding expressions, and simplifying the result. Each step in the process is crucial, requiring careful attention to detail and a solid understanding of algebraic principles. By breaking down the problem into manageable parts, we have demonstrated a systematic approach that can be applied to a variety of similar mathematical challenges. This comprehensive guide serves not only as a solution to the specific problem but also as a valuable resource for enhancing your algebraic skills. Remember, practice and a clear understanding of the fundamentals are the keys to success in mathematics. We hope this detailed explanation has been helpful and has empowered you to tackle algebraic expressions with confidence.