Solving For Z A Step-by-Step Guide To (z-6)/(2z+1) = 10

In the realm of mathematics, solving for variables is a fundamental skill. This article delves into the process of solving the equation $\frac{z-6}{2z+1}=10$ for the variable z. We will embark on a step-by-step journey, meticulously unraveling the equation to arrive at the solution. This exploration will not only provide the answer but also illuminate the underlying principles of algebraic manipulation, empowering you to tackle similar problems with confidence. This article will guide you through the process of finding the value of z that satisfies the given equation. Understanding how to solve for variables like z is crucial in various fields, including algebra, calculus, and physics. By mastering this skill, you'll be well-equipped to tackle more complex mathematical challenges. This equation might seem daunting at first, but with a systematic approach, we can break it down into manageable steps. Our goal is to isolate z on one side of the equation, revealing its value. This involves using algebraic operations to manipulate the equation while maintaining its balance. We'll explore each step in detail, ensuring a clear understanding of the process. Whether you're a student learning algebra or someone looking to refresh your math skills, this article provides a comprehensive guide to solving for z in this specific equation. We'll cover the essential concepts and techniques, making the solution accessible to everyone. Let's embark on this mathematical journey together and unravel the value of z.

Step 1: Eliminate the Fraction

The first hurdle in solving the equation $\frac{z-6}{2z+1}=10$ is the fraction. To eliminate it, we multiply both sides of the equation by the denominator, which is $(2z+1)$. This is a crucial step because it transforms the equation into a more manageable form, free from fractions. Multiplying both sides by the same expression maintains the equality, a fundamental principle of algebraic manipulation. This operation effectively "cancels out" the denominator on the left side, leaving us with a simpler equation. Remember, the goal is to isolate z, and eliminating the fraction is a significant step in that direction. This process is not just about mechanics; it's about understanding the underlying principle of maintaining balance in an equation. By multiplying both sides by the same factor, we ensure that the equation remains true. Now, let's delve into the result of this multiplication and see how it simplifies the equation further. This step sets the stage for the subsequent steps, where we will continue to manipulate the equation to isolate z. It's like laying the foundation for a building; a solid first step is crucial for the success of the entire process. As we move forward, you'll see how each step builds upon the previous one, leading us closer to the solution. So, let's embrace this initial step and proceed with confidence, knowing that we are on the right path to solving for z.

Step 2: Distribute on Both Sides

After eliminating the fraction, the equation becomes $z-6 = 10(2z+1)$. The next step involves distributing the 10 on the right side of the equation. This means multiplying 10 by each term inside the parentheses, which are $2z$ and $1$. Distribution is a fundamental property in algebra that allows us to simplify expressions involving parentheses. It ensures that each term inside the parentheses is properly accounted for when multiplied by the factor outside. This step is crucial because it expands the equation, making it easier to combine like terms and isolate z. By distributing the 10, we transform the equation into a more workable form, where we can start grouping the z terms together. This process is like unraveling a complex knot; each step brings us closer to a clearer picture. The distributed equation will have z terms on both sides, which we will then consolidate in the next step. Understanding distribution is not just about applying a rule; it's about grasping the underlying concept of how multiplication interacts with addition and subtraction. It's a cornerstone of algebraic manipulation, and mastering it will greatly enhance your problem-solving abilities. So, let's carefully perform the distribution, ensuring that each term is multiplied correctly, and prepare ourselves for the next stage of isolating z. This step is a bridge between the initial equation and the simplified form that will ultimately reveal the value of z.

Step 3: Combine Like Terms

Following the distribution, we have the equation $z-6 = 20z + 10$. Now, our objective is to combine like terms. This involves gathering all the z terms on one side of the equation and all the constant terms on the other side. This process simplifies the equation and brings us closer to isolating z. Combining like terms is a core algebraic technique that involves adding or subtracting terms with the same variable and exponent. In this case, we have z terms and constant terms. To achieve this, we can subtract z from both sides and subtract 10 from both sides. This maintains the equality of the equation while strategically moving the terms around. The goal is to have all the z terms on one side and the numerical values on the other. This step is like organizing a messy room; we're grouping similar items together to create a more orderly environment. By combining like terms, we reduce the complexity of the equation and make it easier to solve for z. This process requires careful attention to signs and operations, ensuring that each term is moved and combined correctly. Understanding how to combine like terms is essential for solving various algebraic equations and simplifying expressions. It's a fundamental skill that forms the basis for more advanced mathematical concepts. So, let's meticulously combine the terms, ensuring that we maintain the balance of the equation and pave the way for the final step of isolating z.

Step 4: Isolate z

After combining like terms, the equation is $-16 = 19z$. The final step is to isolate z. This means getting z by itself on one side of the equation. To achieve this, we divide both sides of the equation by the coefficient of z, which is 19. Isolating the variable is the ultimate goal in solving any algebraic equation. It's the process of undoing all the operations that are being performed on the variable until it stands alone, revealing its value. Dividing both sides by the same non-zero number maintains the equality of the equation, a fundamental principle of algebra. This operation effectively "cancels out" the multiplication by 19 on the right side, leaving us with z isolated. The result is the value of z that satisfies the original equation. This step is like the final piece of a puzzle; once it's in place, the solution is revealed. Isolating the variable requires a clear understanding of inverse operations and how they can be used to undo the operations being performed on the variable. This is a critical skill in algebra and is essential for solving a wide range of equations. So, let's perform the division carefully, ensuring that we obtain the correct value of z. This final step is the culmination of all our efforts, bringing us to the solution we've been seeking. By mastering this technique, you'll be able to confidently solve for variables in various mathematical contexts.

Solution

Performing the division, we find that $z = \frac{-16}{19}$. This is the solution to the equation $\frac{z-6}{2z+1}=10$. We can verify this solution by substituting it back into the original equation and checking if it holds true. The solution $z = \frac{-16}{19}$ represents the value of z that makes the equation a true statement. It's the answer we've been working towards throughout this process. This solution demonstrates the power of algebraic manipulation in solving equations. By systematically applying the principles of algebra, we were able to isolate z and determine its value. This process is not just about finding the answer; it's about understanding the underlying mathematical concepts and developing problem-solving skills. Verifying the solution is a crucial step in the problem-solving process. It ensures that we haven't made any errors along the way and that our solution is indeed correct. By substituting the value of z back into the original equation, we can confirm that both sides of the equation are equal, thus validating our solution. This solution provides a sense of accomplishment and reinforces the understanding of the algebraic techniques used. It's a testament to the power of mathematics in solving real-world problems and challenges. So, let's celebrate our success in finding the value of z and appreciate the journey we've taken to get here. This solution is not just a number; it's a symbol of our mathematical prowess and problem-solving abilities.

Conclusion

In conclusion, we have successfully solved the equation $\frac{z-6}{2z+1}=10$ for z, finding that $z = \frac{-16}{19}$. This journey has not only provided us with the solution but has also reinforced our understanding of fundamental algebraic principles. Solving for variables is a cornerstone of mathematics, and the techniques we've employed here are applicable to a wide range of problems. From eliminating fractions to combining like terms and isolating the variable, each step has been crucial in arriving at the solution. The process of solving this equation highlights the importance of a systematic and methodical approach to problem-solving. By breaking down the problem into smaller, manageable steps, we were able to tackle it effectively and confidently. This approach is not just applicable to mathematics; it's a valuable skill in all aspects of life. The solution we've found is not just a number; it's a testament to our ability to think critically, apply mathematical principles, and persevere through challenges. It's a symbol of our problem-solving prowess and our understanding of the language of mathematics. As we move forward in our mathematical journey, let's carry with us the lessons learned from this experience. Let's continue to embrace the power of algebraic manipulation and apply it to solve even more complex and challenging problems. This equation may be just one small step in our mathematical exploration, but it's a step that has strengthened our foundation and prepared us for future endeavors. So, let's celebrate our success and continue to explore the fascinating world of mathematics with curiosity and enthusiasm.