Solving Y - 1/2 = -1/7 A Step-by-Step Guide

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Introduction

In this comprehensive guide, we will delve into the process of solving the equation y - 1/2 = -1/7. This is a fundamental problem in algebra, and mastering the techniques to solve it is crucial for understanding more complex mathematical concepts. Our approach will be methodical, ensuring that each step is clearly explained and easy to follow. Whether you're a student grappling with algebraic equations for the first time or someone looking to refresh your skills, this guide will provide you with the tools and understanding you need. We will explore the underlying principles of equation solving, emphasizing the importance of maintaining balance and performing operations that preserve the equality. By the end of this article, you will not only be able to solve this specific equation but also apply these techniques to a wide range of similar problems. So, let's embark on this mathematical journey and unlock the solution to y - 1/2 = -1/7.

Understanding the Equation

Before diving into the solution, it's essential to understand the anatomy of the equation y - 1/2 = -1/7. This equation is a linear equation in one variable, where 'y' represents the unknown value we aim to find. The equation states that when 1/2 is subtracted from 'y', the result is -1/7. To solve for 'y', we need to isolate it on one side of the equation. This involves performing operations that maintain the equality, meaning whatever we do to one side of the equation, we must also do to the other side. This principle is the cornerstone of equation solving and ensures that the solution we arrive at is accurate. The fractions in the equation, 1/2 and -1/7, add a layer of complexity, but with a clear understanding of fraction arithmetic, we can easily navigate this. Remember, the goal is to find the value of 'y' that makes the equation true, and by systematically isolating 'y', we can achieve this.

Step 1: Isolate 'y'

The primary goal in solving any algebraic equation is to isolate the variable. In the equation y - 1/2 = -1/7, 'y' is currently being subtracted by 1/2. To isolate 'y', we need to perform the inverse operation, which is adding 1/2 to both sides of the equation. This step is crucial because it maintains the balance of the equation. Adding 1/2 to the left side cancels out the -1/2, leaving 'y' by itself. However, we must also add 1/2 to the right side to preserve the equality. This can be represented mathematically as: y - 1/2 + 1/2 = -1/7 + 1/2. By adding 1/2 to both sides, we ensure that the equation remains balanced, and we move closer to finding the value of 'y'. This step highlights the fundamental principle of equation solving: performing the same operation on both sides to maintain equality.

Step 2: Adding Fractions

Now that we have isolated 'y', our equation looks like this: y = -1/7 + 1/2. The next step is to add the fractions on the right side of the equation. To add fractions, they must have a common denominator. The denominators in our fractions are 7 and 2. The least common multiple (LCM) of 7 and 2 is 14. Therefore, we need to convert both fractions to have a denominator of 14. To convert -1/7 to a fraction with a denominator of 14, we multiply both the numerator and the denominator by 2: (-1 * 2) / (7 * 2) = -2/14. Similarly, to convert 1/2 to a fraction with a denominator of 14, we multiply both the numerator and the denominator by 7: (1 * 7) / (2 * 7) = 7/14. Now, we can rewrite our equation as: y = -2/14 + 7/14. With the fractions having a common denominator, we can proceed to add them.

Step 3: Simplifying the Result

With a common denominator, we can now add the fractions: y = -2/14 + 7/14. To add fractions with the same denominator, we simply add the numerators and keep the denominator the same. So, -2 + 7 = 5. Therefore, y = 5/14. This is the solution to the equation. The fraction 5/14 is in its simplest form because 5 and 14 have no common factors other than 1. This means we cannot reduce the fraction further. The solution y = 5/14 tells us that if we substitute 5/14 for 'y' in the original equation, the equation will be true. It's always a good practice to check your solution by substituting it back into the original equation to ensure accuracy. In this case, substituting 5/14 for 'y' in y - 1/2 = -1/7 should yield a true statement.

Verification of the Solution

To ensure the accuracy of our solution, it's crucial to verify it. This involves substituting the value we found for 'y' back into the original equation. Our solution is y = 5/14, and the original equation is y - 1/2 = -1/7. Substituting 5/14 for 'y', we get: 5/14 - 1/2 = -1/7. Now, we need to check if this statement is true. To subtract the fractions on the left side, we need a common denominator, which we already determined to be 14. We convert 1/2 to 7/14, so the equation becomes: 5/14 - 7/14 = -1/7. Subtracting the fractions on the left side, we get: (5 - 7) / 14 = -2/14. Simplifying -2/14, we divide both the numerator and the denominator by 2, which gives us -1/7. Therefore, the equation becomes: -1/7 = -1/7. This statement is true, which confirms that our solution y = 5/14 is correct. Verifying the solution is an essential step in problem-solving, as it helps catch any potential errors and ensures confidence in the answer.

Conclusion

In conclusion, we have successfully solved the equation y - 1/2 = -1/7 by systematically isolating the variable 'y' and performing the necessary arithmetic operations. Our step-by-step approach involved adding 1/2 to both sides of the equation, finding a common denominator to add fractions, simplifying the result, and verifying the solution. The final answer is y = 5/14. This process not only provides the solution to this specific equation but also illustrates the fundamental principles of equation solving in algebra. The key takeaways from this exercise are the importance of maintaining balance in an equation by performing the same operations on both sides, the ability to work with fractions, and the necessity of verifying the solution. By mastering these techniques, you can confidently tackle a wide range of algebraic equations and further your understanding of mathematics. Remember, practice is key to proficiency, so continue to apply these methods to various problems to solidify your skills.