Solving The Equation -1/4x + 1/2 = 1/8x - 1/3 A Step-by-Step Guide

Introduction

In the realm of mathematics, solving equations is a fundamental skill. It is the cornerstone of algebra and essential for various scientific and engineering applications. This article aims to provide a detailed, step-by-step guide on how to solve the equation $-\frac{1}{4}x + \frac{1}{2} = \frac{1}{8}x - \frac{1}{3}$. We will explore the underlying principles, the techniques involved, and the reasoning behind each step. Whether you are a student grappling with algebra or someone looking to brush up on your math skills, this guide will provide you with a comprehensive understanding of the process. So, let's dive in and unravel the intricacies of this equation!

Understanding the Basics of Algebraic Equations

Before we delve into the specifics of solving this particular equation, it's crucial to understand the basic principles of algebraic equations. An equation is a mathematical statement that asserts the equality of two expressions. These expressions are connected by an equals sign (=). The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true. In simpler terms, we are looking for the value of 'x' that, when substituted into the equation, will make both sides of the equation equal. Equations can be linear, quadratic, or of higher degrees, depending on the highest power of the variable involved. Our equation, $-\frac{1}{4}x + \frac{1}{2} = \frac{1}{8}x - \frac{1}{3}$, is a linear equation because the highest power of 'x' is 1. Solving linear equations generally involves isolating the variable on one side of the equation by performing the same operations on both sides. This principle of maintaining equality is the bedrock of solving any algebraic equation. Remember, whatever you do to one side of the equation, you must do to the other side to keep the equation balanced. This includes addition, subtraction, multiplication, and division. With this foundational understanding, we are now ready to tackle the given equation step by step.

Step-by-Step Solution

1. Eliminate Fractions

The first step in solving the equation $-\frac{1}{4}x + \frac{1}{2} = \frac{1}{8}x - \frac{1}{3}$ is to eliminate the fractions. Fractions can make the equation look more complicated and increase the chances of making a mistake. To get rid of the fractions, we need to find the least common multiple (LCM) of the denominators. The denominators in our equation are 4, 2, 8, and 3. The LCM of these numbers is 24. Now, we multiply both sides of the equation by 24. This will clear out all the fractions. Multiplying the left side by 24, we get: $24 * (-\frac{1}{4}x + \frac{1}{2}) = 24 * (-\frac{1}{4}x) + 24 * (\frac{1}{2}) = -6x + 12$. Multiplying the right side by 24, we get: $24 * (\frac{1}{8}x - \frac{1}{3}) = 24 * (\frac{1}{8}x) - 24 * (\frac{1}{3}) = 3x - 8$. So, our equation now becomes: -6x + 12 = 3x - 8. This transformed equation is much easier to work with than the original equation with fractions. By eliminating the fractions, we have simplified the equation and made it more manageable for the subsequent steps. This technique is a crucial step in solving any equation involving fractions, and mastering it will significantly enhance your problem-solving skills in algebra.

2. Combine Like Terms

The next step in solving the equation -6x + 12 = 3x - 8 is to combine like terms. Combining like terms involves grouping the terms with the variable 'x' on one side of the equation and the constant terms on the other side. This step helps to isolate the variable and move closer to the solution. To do this, we can add 6x to both sides of the equation. This will eliminate the '-6x' term on the left side and move the 'x' term to the right side. Adding 6x to both sides, we get: -6x + 12 + 6x = 3x - 8 + 6x, which simplifies to 12 = 9x - 8. Now, we need to move the constant term '-8' from the right side to the left side. To do this, we add 8 to both sides of the equation. Adding 8 to both sides, we get: 12 + 8 = 9x - 8 + 8, which simplifies to 20 = 9x. At this point, we have successfully combined like terms, and the equation is now in a simpler form with the 'x' term isolated on one side and the constant term on the other side. This step is crucial for simplifying equations and making them easier to solve. By mastering the technique of combining like terms, you can effectively streamline your problem-solving process in algebra and tackle more complex equations with confidence.

3. Isolate the Variable

Now that we have the equation 20 = 9x, the next step is to isolate the variable 'x'. Isolating the variable means getting 'x' by itself on one side of the equation. To do this, we need to undo the operation that is being performed on 'x'. In this case, 'x' is being multiplied by 9. To undo multiplication, we perform the inverse operation, which is division. Therefore, we need to divide both sides of the equation by 9. Dividing both sides by 9, we get: 20 / 9 = (9x) / 9. This simplifies to x = 20/9. At this point, we have successfully isolated the variable 'x' and found its value. The value of 'x' that satisfies the equation is 20/9. This fraction cannot be simplified further, so it is the final answer. Isolating the variable is the ultimate goal in solving any equation, and this step demonstrates the fundamental principle of using inverse operations to unravel the equation and find the value of the unknown. By understanding and applying this technique, you can confidently solve a wide range of algebraic equations.

4. Verify the Solution

The final step in solving any equation is to verify the solution. Verifying the solution involves substituting the value we found for 'x' back into the original equation to ensure that it makes the equation true. This step is crucial because it helps to catch any errors that may have occurred during the solving process. Our original equation was: $-\frac{1}{4}x + \frac{1}{2} = \frac{1}{8}x - \frac{1}{3}$. We found that x = 20/9. Now, we substitute x = 20/9 into the original equation: $-\frac{1}{4} * (20/9) + \frac{1}{2} = \frac{1}{8} * (20/9) - \frac{1}{3}$. Simplifying the left side, we get: -5/9 + 1/2 = (-10 + 9) / 18 = -1/18. Simplifying the right side, we get: 5/18 - 1/3 = (5 - 6) / 18 = -1/18. Since both sides of the equation are equal (-1/18 = -1/18), our solution x = 20/9 is correct. Verification is a critical step in the problem-solving process. It not only confirms the accuracy of your solution but also reinforces your understanding of the equation and the steps involved in solving it. By making verification a habit, you can improve your problem-solving skills and minimize the chances of errors.

Final Answer

After following all the steps outlined above, we have successfully solved the equation $-\frac{1}{4}x + \frac{1}{2} = \frac{1}{8}x - \frac{1}{3}$. By eliminating fractions, combining like terms, isolating the variable, and verifying the solution, we have determined that: x = 20/9. This is the value of 'x' that makes the equation true. Solving equations is a fundamental skill in mathematics, and this comprehensive guide has provided a detailed explanation of the process involved. By mastering these steps and practicing regularly, you can confidently tackle a wide range of algebraic equations and enhance your problem-solving abilities.

Conclusion

In conclusion, solving the equation $-\frac{1}{4}x + \frac{1}{2} = \frac{1}{8}x - \frac{1}{3}$ demonstrates the fundamental principles of algebra. We began by eliminating fractions to simplify the equation, then combined like terms to isolate the variable 'x'. The process of isolating 'x' involved using inverse operations, a cornerstone of algebraic manipulation. Finally, we verified our solution by substituting it back into the original equation, ensuring accuracy. This step-by-step approach not only provides the solution x = 20/9 but also reinforces the importance of each step in the problem-solving process. Mastering these techniques is crucial for success in algebra and beyond. Equations are the language of mathematics, and the ability to solve them is essential for understanding and applying mathematical concepts in various fields. By practicing and applying these methods, you can develop a strong foundation in algebra and confidently approach more complex mathematical challenges. Remember, the key to success in mathematics is consistent practice and a clear understanding of the underlying principles.