Solving For X In The Equation 2/3 = 8/(x+6) A Step-by-Step Guide
In the realm of mathematics, solving equations is a fundamental skill. This article delves into a step-by-step solution for the equation 2/3 = 8/(x+6), offering a clear and concise explanation that will benefit students and math enthusiasts alike. Understanding how to solve for variables like 'x' is crucial for various mathematical applications, including algebra, calculus, and beyond. Let's embark on this mathematical journey and unravel the value of 'x'. This comprehensive guide will not only provide the solution but also the underlying principles and techniques involved. Mastering these concepts is essential for tackling more complex mathematical problems. We aim to make this process as straightforward and understandable as possible. Our focus is on building a solid foundation in algebraic problem-solving. So, let's dive in and explore the world of equations!
Understanding the Problem
Before diving into the solution, let's first understand the problem at hand. We are given the equation 2/3 = 8/(x+6), which is a proportion. A proportion is an equation stating that two ratios (or fractions) are equal. Our goal is to find the value of 'x' that makes this equation true. In essence, we need to isolate 'x' on one side of the equation. This involves using algebraic manipulations to simplify the equation and ultimately solve for 'x'. Understanding the structure of the equation is the first step towards finding the solution. We need to identify the operations being performed on 'x' and then apply inverse operations to undo them. This process is the cornerstone of solving algebraic equations. The key is to maintain balance; whatever operation we perform on one side of the equation, we must also perform on the other side. This ensures that the equality remains valid. With a clear understanding of the problem, we can now proceed to the solution.
Step-by-Step Solution
Now, let's break down the step-by-step solution to the equation 2/3 = 8/(x+6). The most common method for solving proportions is cross-multiplication. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. In our case, this means multiplying 2 by (x+6) and 8 by 3. This gives us the equation 2(x+6) = 8 * 3. Next, we need to simplify both sides of the equation. On the left side, we distribute the 2 across the parentheses: 2 * x + 2 * 6, which simplifies to 2x + 12. On the right side, 8 * 3 equals 24. So, our equation now looks like this: 2x + 12 = 24. To isolate the term with 'x', we subtract 12 from both sides of the equation: 2x + 12 - 12 = 24 - 12, which simplifies to 2x = 12. Finally, to solve for 'x', we divide both sides of the equation by 2: 2x / 2 = 12 / 2, which gives us x = 6. Therefore, the solution to the equation is x = 6. This step-by-step approach ensures clarity and accuracy in the solution process. Each step is a logical progression, building upon the previous one. By following these steps, you can confidently solve similar proportion problems.
Verification of the Solution
After obtaining a solution, it's always a good practice to verify its correctness. This ensures that our answer is accurate and satisfies the original equation. To verify our solution, we substitute x = 6 back into the original equation: 2/3 = 8/(x+6). Replacing 'x' with 6, we get 2/3 = 8/(6+6), which simplifies to 2/3 = 8/12. Now, we can simplify the fraction 8/12 by dividing both the numerator and the denominator by their greatest common divisor, which is 4. This gives us 8/12 = (8/4)/(12/4) = 2/3. So, our equation becomes 2/3 = 2/3, which is a true statement. This confirms that our solution, x = 6, is indeed correct. Verification is a crucial step in problem-solving, as it helps identify any potential errors made during the process. By substituting the solution back into the original equation, we can ensure the accuracy of our answer. This practice builds confidence and reinforces understanding of the concepts involved.
Alternative Methods
While cross-multiplication is a common method for solving proportions, there are alternative approaches we can explore. One such method involves manipulating the fractions to have a common denominator or a common numerator. Let's revisit our equation: 2/3 = 8/(x+6). Notice that the numerator on the right side (8) is four times the numerator on the left side (2). To make the numerators equal, we could multiply the entire fraction on the left side by 4/4 (which is equal to 1, so it doesn't change the value of the fraction). This gives us (2/3) * (4/4) = 8/12. Now, our equation looks like this: 8/12 = 8/(x+6). Since the numerators are equal, the denominators must also be equal for the equation to hold true. Therefore, we can set 12 equal to (x+6): 12 = x + 6. Subtracting 6 from both sides, we get x = 6, which is the same solution we obtained using cross-multiplication. This alternative method demonstrates that there can be multiple pathways to arrive at the correct answer. Exploring different methods enhances problem-solving skills and provides a deeper understanding of mathematical concepts. It also allows you to choose the method that you find most efficient and comfortable. Another approach could involve taking the reciprocal of both sides of the equation and then solving for 'x'. This can sometimes simplify the equation and make it easier to solve.
Common Mistakes to Avoid
When solving equations, it's essential to be aware of common mistakes that students often make. Identifying and avoiding these pitfalls can significantly improve accuracy and problem-solving skills. One common mistake is incorrectly applying the distributive property. For example, in the equation 2(x+6) = 24, some students might mistakenly multiply only the 'x' by 2, resulting in 2x + 6 = 24, instead of the correct 2x + 12 = 24. Another frequent error is failing to perform the same operation on both sides of the equation. Remember, to maintain balance, any operation applied to one side must also be applied to the other side. For instance, when solving 2x + 12 = 24, if we subtract 12 from the left side, we must also subtract 12 from the right side. A third common mistake is incorrectly simplifying fractions or performing arithmetic operations. Always double-check your calculations to ensure accuracy. Additionally, it's crucial to verify the solution by substituting it back into the original equation. This helps catch any errors made during the solving process. Being mindful of these common mistakes and actively working to avoid them will lead to greater success in solving equations. Practice and attention to detail are key to mastering these skills. Furthermore, understanding the underlying principles of algebra is crucial for avoiding errors and solving equations effectively.
Practice Problems
To solidify your understanding of solving equations, let's work through some practice problems. These problems will help you apply the techniques we've discussed and build confidence in your abilities.
Problem 1: Solve for y in the equation 3/4 = 9/(y+5)
Problem 2: Solve for z in the equation 5/2 = 15/(z-3)
Problem 3: Solve for a in the equation 1/5 = 4/(a+7)
Problem 4: Solve for b in the equation 7/3 = 21/(b-2)
Problem 5: Solve for c in the equation 4/6 = 10/(c+1)
These practice problems offer a range of scenarios to test your skills. Remember to follow the step-by-step approach we outlined earlier: cross-multiply, simplify, isolate the variable, and verify your solution. Working through these problems will not only reinforce your understanding but also help you develop fluency in solving equations. Don't be afraid to make mistakes; they are opportunities for learning and growth. If you encounter difficulties, revisit the step-by-step solution and alternative methods discussed in this article. The key is consistent practice and a willingness to learn from your errors. Furthermore, consider exploring additional resources, such as textbooks, online tutorials, and practice worksheets, to further enhance your problem-solving abilities.
Conclusion
In conclusion, solving for 'x' in the equation 2/3 = 8/(x+6) is a fundamental algebraic skill. We've explored a step-by-step solution, including cross-multiplication, simplification, and verification. We've also discussed alternative methods and common mistakes to avoid. By understanding the principles and techniques outlined in this article, you can confidently tackle similar equations. Remember, practice is key to mastering any mathematical concept. The more you practice, the more comfortable and proficient you will become. The ability to solve equations is not only essential for success in mathematics but also for various real-world applications. From calculating finances to designing structures, equations are used extensively in various fields. Therefore, investing time and effort in developing these skills is a worthwhile endeavor. We encourage you to continue exploring the world of mathematics and challenging yourself with new problems. With dedication and perseverance, you can achieve your mathematical goals. This comprehensive guide has provided you with the tools and knowledge necessary to solve for 'x' and beyond. Keep practicing, keep learning, and keep exploring the fascinating world of mathematics!
