Solving The Equation -1/2(-10x - 10) - 8 = 4(3x - 2) - 9 A Step-by-Step Guide

Introduction

In the realm of mathematics, solving linear equations is a fundamental skill. Linear equations, characterized by variables raised to the power of one, appear in various contexts, from basic algebra to advanced calculus and real-world applications. This article provides a comprehensive, step-by-step guide to solving the linear equation $-\frac{1}{2}(-10x - 10) - 8 = 4(3x - 2) - 9$. We will break down each stage, ensuring clarity and understanding, making this a valuable resource for students, educators, and anyone seeking to enhance their problem-solving abilities. Our exploration will not only focus on the mechanics of solving the equation but also delve into the underlying principles that make these methods effective. By mastering the techniques outlined here, you'll be well-equipped to tackle a wide array of algebraic challenges. This article serves as an invaluable resource for students, educators, and anyone looking to sharpen their mathematical skills. Our approach emphasizes clarity and a step-by-step methodology, ensuring that even those new to algebra can follow along and grasp the concepts. We'll begin with the basics, gradually building up to more complex steps, so you'll feel confident and capable at every stage of the process. From simplifying expressions to isolating variables, each technique will be thoroughly explained and illustrated. By the end of this article, you'll have a solid understanding of how to solve this specific equation and, more importantly, how to apply these skills to solve similar problems in the future. Whether you're studying for an exam, brushing up on your algebra, or simply enjoy the intellectual challenge of solving equations, this guide is designed to meet your needs and elevate your mathematical proficiency. Let's embark on this journey together, unraveling the mysteries of linear equations and discovering the power of algebraic problem-solving.

Step 1: Distribute and Simplify Both Sides

The initial step in solving this equation involves distributing the constants on both sides to eliminate parentheses. For the left side, we have $-\frac{1}{2}(-10x - 10)$. Distributing $-\frac{1}{2}$ across the terms inside the parentheses gives us:

$-\frac{1}{2} * -10x = 5x$ and $-\frac{1}{2} * -10 = 5$

So, the left side becomes $5x + 5 - 8$. Combining the constants, we get $5x - 3$. On the right side, we have $4(3x - 2) - 9$. Distributing the 4 across the terms inside the parentheses yields:

$4 * 3x = 12x$ and $4 * -2 = -8$

Thus, the right side becomes $12x - 8 - 9$. Combining the constants, we get $12x - 17$. After this crucial distribution and simplification, the equation is now much cleaner and easier to manage. We have transformed the original equation into a more streamlined form, setting the stage for the next steps in our solution process. By carefully applying the distributive property and combining like terms, we have effectively reduced the complexity of the equation, making it more accessible and less intimidating. This initial simplification is a key strategy in solving any algebraic equation, as it helps to isolate the variables and constants, paving the way for a straightforward solution. The ability to efficiently distribute and simplify expressions is a fundamental skill in algebra, and mastering this step will greatly enhance your problem-solving capabilities. As we move forward, we will continue to employ similar techniques to further refine the equation and ultimately arrive at the solution. Remember, the key to success in algebra lies in understanding each step and applying it with precision and care. With the equation now simplified, we are well-positioned to proceed to the next stage and work towards finding the value of x.

Step 2: Isolate the Variable Terms

In this step, the primary goal is to isolate the variable terms on one side of the equation. Having simplified both sides in the previous step, our equation now stands as $5x - 3 = 12x - 17$. To isolate the variable terms, we want to gather all terms containing x on one side and all constant terms on the other. A common approach is to move the term with the smaller coefficient of x to the side with the larger coefficient. In this case, we have 5x on the left and 12x on the right. To move the 5x term, we subtract 5x from both sides of the equation. This maintains the balance of the equation while effectively eliminating the x term from the left side. Subtracting 5x from both sides gives us: $5x - 3 - 5x = 12x - 17 - 5x$. Simplifying this, we get $-3 = 7x - 17$. Now, all the x terms are on the right side, and we can proceed to isolate the constant terms. This systematic approach to isolating variables is a fundamental technique in algebra. By strategically adding or subtracting terms from both sides of the equation, we can rearrange the equation to suit our needs. The key is to always perform the same operation on both sides to maintain equality. This ensures that the equation remains balanced and the solution remains valid. Isolating the variable terms is a critical step in solving any algebraic equation, as it brings us closer to the final solution. By concentrating all the variable terms on one side, we can then focus on isolating the variable itself, which is our ultimate objective. The process of isolating variables may seem complex at first, but with practice, it becomes second nature. By understanding the underlying principles and applying them consistently, you can master this essential skill and confidently tackle a wide range of algebraic problems. As we continue to solve this equation, we will see how each step builds upon the previous one, gradually leading us to the solution. With the variable terms now isolated, we are ready to move on to the next stage and further simplify the equation.

Step 3: Isolate the Constant Terms

After successfully isolating the variable terms on one side of the equation, the next crucial step is to isolate the constant terms on the opposite side. This process will further simplify the equation and bring us closer to the final solution. From the previous step, we have the equation $-3 = 7x - 17$. Our goal now is to isolate the x term by moving the constant term (-17) to the left side of the equation. To do this, we add 17 to both sides of the equation. This operation maintains the equality of the equation while effectively eliminating the constant term from the right side. Adding 17 to both sides gives us: $-3 + 17 = 7x - 17 + 17$. Simplifying this, we get $14 = 7x$. Now, we have successfully isolated the constant terms on the left side and the variable term on the right side. This rearrangement of the equation is a significant step towards solving for x. By isolating the constant terms, we have created a simpler equation that is easier to manipulate and solve. This strategic approach to equation solving is a hallmark of algebraic thinking. By systematically isolating terms, we can break down complex equations into more manageable parts and ultimately arrive at the solution. The process of isolating constant terms is not only essential for solving linear equations but also for solving more complex algebraic problems. By mastering this technique, you will be well-equipped to tackle a wide range of mathematical challenges. As we continue to solve this equation, we will see how each step builds upon the previous one, gradually leading us to the solution. With the constant terms now isolated, we are ready to move on to the final stage and solve for the value of x. The equation $14 = 7x$ is now in a simple form, making it straightforward to determine the value of the variable. Let's proceed to the next step and complete the solution.

Step 4: Solve for x

With the equation now simplified to $14 = 7x$, the final step is to solve for x. This involves isolating x by dividing both sides of the equation by the coefficient of x. In this case, the coefficient of x is 7. To isolate x, we divide both sides of the equation by 7. This operation maintains the equality of the equation while effectively solving for x. Dividing both sides by 7 gives us: $\frac{14}{7} = \frac{7x}{7}$. Simplifying this, we get $2 = x$. Therefore, the solution to the equation is x = 2. This is the final answer, the value of x that satisfies the original equation. The process of solving for x is the culmination of all the previous steps. By carefully distributing, simplifying, and isolating terms, we have successfully transformed the original equation into a form where the value of x can be easily determined. This systematic approach to equation solving is a powerful tool in algebra. By understanding each step and applying it with precision, you can confidently solve a wide range of linear equations. The ability to solve for variables is a fundamental skill in mathematics and is essential for many applications in science, engineering, and other fields. By mastering this technique, you will be well-equipped to tackle a variety of problems and challenges. As we have seen in this example, solving a linear equation involves a series of logical steps, each building upon the previous one. By carefully following these steps, we can break down complex equations into manageable parts and ultimately arrive at the solution. The final solution, x = 2, represents the value that makes the equation true. To verify this solution, we can substitute x = 2 back into the original equation and check if both sides are equal. This is a useful way to ensure that the solution is correct and to reinforce the understanding of the equation-solving process.

Conclusion

In conclusion, we have successfully solved the linear equation $-\frac{1}{2}(-10x - 10) - 8 = 4(3x - 2) - 9$ through a series of carefully executed steps. The solution we arrived at is x = 2. This journey through the equation-solving process has highlighted the importance of several key algebraic techniques. We began by distributing constants to eliminate parentheses, a critical step in simplifying complex expressions. This allowed us to transform the original equation into a more manageable form. Next, we focused on simplifying both sides of the equation by combining like terms. This involved adding or subtracting constants and variable terms to consolidate the equation and make it easier to work with. The process of isolating variable terms was another crucial step. By strategically adding or subtracting terms from both sides of the equation, we were able to gather all the x terms on one side and the constant terms on the other. This laid the foundation for solving for x. Isolating constant terms followed a similar approach, where we moved all the constant terms to the opposite side of the equation from the variable terms. This further simplified the equation and brought us closer to the final solution. Finally, we solved for x by dividing both sides of the equation by the coefficient of x. This gave us the value of x that satisfies the equation. Throughout this process, we emphasized the importance of maintaining the balance of the equation by performing the same operations on both sides. This ensures that the equation remains equal and the solution remains valid. The skills and techniques we have explored in this article are not only applicable to this specific equation but also to a wide range of algebraic problems. By mastering these concepts, you will be well-equipped to tackle more complex equations and mathematical challenges. The ability to solve linear equations is a fundamental skill in mathematics and is essential for many applications in science, engineering, and other fields. We encourage you to practice these techniques and apply them to other problems to further enhance your understanding and problem-solving abilities. The journey of solving this equation has been a valuable learning experience, and we hope that this detailed guide has provided you with the knowledge and confidence to tackle similar problems in the future. Remember, the key to success in mathematics is practice, patience, and a systematic approach.