Solving For U In The Equation 2 = 2(u + 4) - 4u A Step-by-Step Guide

In this article, we will delve into the process of solving for the variable u in the algebraic equation 2 = 2(u + 4) - 4u. This type of problem is fundamental in algebra and involves the application of several key mathematical principles, including the distributive property, combining like terms, and isolating the variable. Our goal is to simplify the equation step-by-step, ensuring a clear and understandable solution for anyone seeking to master basic algebraic techniques. We will break down each step with detailed explanations, making it easy to follow along and learn. Whether you are a student learning algebra for the first time or someone looking to refresh your skills, this guide will provide a comprehensive approach to solving this type of equation. The ability to solve for variables is a crucial skill in various fields, including mathematics, science, engineering, and economics, making it a valuable tool in both academic and professional settings. Understanding the methodology behind solving equations not only helps in finding the correct answer but also in developing problem-solving skills that can be applied to a wide range of situations. We will also emphasize the importance of checking your solution to ensure accuracy, a practice that is essential in mathematical problem-solving. By the end of this article, you will have a clear understanding of how to solve this particular equation and a solid foundation for tackling similar algebraic problems.

Step-by-Step Solution

1. Distribute the 2 across the parentheses:

Our initial equation is 2 = 2(u + 4) - 4u. The first step in simplifying this equation involves applying the distributive property. This property states that a(b + c) = ab + ac. In our case, we need to distribute the 2 across the terms inside the parentheses (u + 4). This means we multiply 2 by both u and 4. This process is crucial because it eliminates the parentheses, allowing us to combine like terms in the subsequent steps. The distributive property is a cornerstone of algebraic manipulation, and mastering it is essential for solving more complex equations. It ensures that each term inside the parentheses is correctly multiplied by the factor outside, maintaining the equation's balance. When we apply the distributive property here, we get 2 * u + 2 * 4, which simplifies to 2u + 8. Substituting this back into our equation, we have 2 = 2u + 8 - 4u. This transformation is a significant step forward in isolating the variable u. By correctly distributing, we've set the stage for further simplification and, ultimately, finding the value of u. Understanding and applying the distributive property correctly is a fundamental skill in algebra and is crucial for solving a wide variety of equations. This step ensures that the equation remains balanced and that we are one step closer to finding the solution.

2. Simplify the equation:

After distributing the 2, our equation becomes 2 = 2u + 8 - 4u. The next step involves simplifying the equation by combining like terms. In this case, the like terms are the terms that contain the variable u: 2u and -4u. Combining like terms is a fundamental algebraic technique that helps to reduce the complexity of the equation and makes it easier to solve. To combine 2u and -4u, we simply add their coefficients: 2 + (-4) = -2. Therefore, 2u - 4u simplifies to -2u. Substituting this back into our equation, we get 2 = -2u + 8. This simplified equation is much easier to work with than the original. It has fewer terms and isolates the variable u on one side of the equation, making it clearer how to proceed. Simplifying equations by combining like terms is a crucial skill in algebra. It allows us to reduce the equation to its simplest form, which is essential for isolating the variable and finding its value. This step demonstrates the importance of organizing and simplifying equations before attempting to solve them, which is a key strategy in mathematical problem-solving. The ability to combine like terms efficiently is a valuable asset in tackling more complex algebraic problems.

3. Subtract 8 from both sides:

Now that we have the simplified equation 2 = -2u + 8, our next goal is to isolate the term containing the variable u. To do this, we need to eliminate the constant term, which is 8, from the right side of the equation. The principle we use here is that we can perform the same operation on both sides of an equation without changing its balance. Therefore, to eliminate 8, we subtract 8 from both sides of the equation. This maintains the equality and moves us closer to isolating u. Subtracting 8 from the left side (2) gives us 2 - 8 = -6. Subtracting 8 from the right side (-2u + 8) gives us -2u + 8 - 8, which simplifies to -2u. So, our equation now becomes -6 = -2u. This is a significant step because it isolates the term with u on one side of the equation, making it easier to solve for u. The act of subtracting 8 from both sides demonstrates a key algebraic technique: using inverse operations to isolate a variable. This principle is fundamental to solving equations and is used extensively in algebra and other mathematical disciplines. By correctly applying this step, we have simplified the equation further and are closer to finding the value of u.

4. Divide both sides by -2:

After subtracting 8 from both sides, our equation is now -6 = -2u. To finally isolate the variable u, we need to get rid of the coefficient -2 that is multiplying u. We accomplish this by performing the inverse operation, which is division. We divide both sides of the equation by -2, maintaining the balance of the equation. This is a crucial step in solving for u because it directly leads to the solution. Dividing -6 by -2 gives us -6 / -2 = 3. Dividing -2u by -2 gives us (-2u) / -2 = u. Therefore, our equation simplifies to 3 = u. This means that the value of u that satisfies the original equation is 3. This step demonstrates the fundamental algebraic principle of using inverse operations to isolate a variable and solve for its value. Dividing both sides of the equation by the same number ensures that the equation remains balanced and that we are correctly determining the value of u. This is a core technique in algebra and is essential for solving a wide range of equations. The result, u = 3, is the solution to our problem.

5. The solution:

After performing all the necessary algebraic steps, we have successfully solved for u in the equation 2 = 2(u + 4) - 4u. Our final result is u = 3. This means that when u is replaced with 3 in the original equation, the equation holds true. To be absolutely sure of our solution, it's always a good practice to check our answer by substituting it back into the original equation. This verification step is crucial because it helps us catch any potential errors made during the solving process. Substituting u = 3 into the original equation, we get: 2 = 2(3 + 4) - 4(3). Simplifying this, we have: 2 = 2(7) - 12, which further simplifies to: 2 = 14 - 12, and finally: 2 = 2. Since the left side of the equation equals the right side, our solution u = 3 is correct. This verification step reinforces the importance of accuracy in algebraic problem-solving. It not only confirms that we have found the correct solution but also enhances our understanding of the equation and the steps involved in solving it. The process of solving for u and then verifying the solution is a complete demonstration of algebraic problem-solving skills.

Conclusion

In this comprehensive guide, we have demonstrated a step-by-step method to solve the equation 2 = 2(u + 4) - 4u for the variable u. We began by distributing the 2 across the parentheses, a crucial step that allowed us to eliminate the parentheses and simplify the equation. We then combined like terms, which further reduced the complexity of the equation and made it easier to isolate the variable. Next, we used inverse operations to isolate u, first by subtracting 8 from both sides and then by dividing both sides by -2. This process ultimately led us to the solution u = 3. To ensure the accuracy of our solution, we verified our answer by substituting it back into the original equation. This verification step confirmed that our solution was indeed correct. The process of solving this equation highlights several key algebraic principles, including the distributive property, combining like terms, and using inverse operations. These principles are fundamental to solving a wide range of algebraic problems and are essential skills for anyone studying mathematics. Furthermore, the importance of checking our solution cannot be overstated, as it provides a safeguard against errors and reinforces our understanding of the equation. By mastering these techniques, you can confidently tackle similar algebraic problems and enhance your mathematical problem-solving abilities.

This step-by-step approach not only provides a clear solution to the equation but also reinforces fundamental algebraic concepts. By understanding these principles, students and anyone interested in mathematics can build a strong foundation for tackling more complex problems. The ability to solve for variables is a valuable skill in various fields, making this a worthwhile exercise in mathematical problem-solving.