Solving For X In The Equation -2/3x - 3/2 = -6/7 A Step-by-Step Guide

In this comprehensive guide, we will delve into the process of solving for x in the equation $-\frac{2}{3}x - \frac{3}{2} = -\frac{6}{7}$. This type of equation, a linear equation in one variable, is a cornerstone of algebra, and mastering its solution is crucial for success in higher-level mathematics. Our discussion will cover each step in detail, ensuring clarity and understanding. We will begin by outlining the fundamental principles of equation solving and then move into the step-by-step method to isolate x, providing explanations and justifications for each operation. The importance of verifying the solution will also be emphasized to ensure accuracy. Whether you are a student learning algebra for the first time or someone looking to refresh your skills, this article is designed to provide a clear and thorough understanding of how to solve for x in this type of equation. By the end, you will not only be able to solve this specific equation but also grasp the underlying concepts that apply to a wide range of algebraic problems. This journey through the solution will equip you with the tools and confidence to tackle similar mathematical challenges, setting a solid foundation for your algebraic endeavors. Remember, practice is key in mastering these skills, so feel free to work through the solution alongside the article, reinforcing your understanding with each step.

Understanding the Basics of Solving Equations

Before we dive into the specifics of solving for x in the given equation, it's essential to establish a solid understanding of the foundational principles that govern equation solving in algebra. At its core, solving an equation involves isolating the variable on one side of the equation to determine its value. This is achieved by performing operations that maintain the balance of the equation, ensuring that both sides remain equal. The golden rule of equation solving is: Whatever operation you perform on one side of the equation, you must perform the same operation on the other side. This principle ensures that the equality is preserved throughout the solution process. There are several fundamental operations we can use to manipulate equations, including addition, subtraction, multiplication, and division. These operations are the building blocks of algebraic problem-solving, allowing us to systematically simplify equations and isolate the variable of interest. Understanding these operations and knowing when and how to apply them is crucial. For instance, if a number is added to the variable, we can subtract that number from both sides to isolate the variable. Similarly, if the variable is multiplied by a number, we can divide both sides by that number. These inverse operations are key to unraveling the equation and revealing the value of x. Moreover, we must consider the order of operations (PEMDAS/BODMAS) when simplifying equations. This mnemonic helps us remember the correct sequence: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Adhering to this order ensures that we perform operations in the correct sequence, preventing errors in our calculations. By mastering these basics, you are laying a strong foundation for solving for x in more complex equations and algebraic problems. The ability to confidently apply these principles will empower you to tackle a wide range of mathematical challenges, reinforcing your understanding and problem-solving skills.

Step-by-Step Solution: Isolating x

Now, let's embark on the step-by-step journey of isolating x in the equation $-\frac{2}{3}x - \frac{3}{2} = -\frac{6}{7}$. This process involves strategically applying algebraic operations to gradually peel away the terms surrounding x until we arrive at its value. Our primary objective is to isolate x on one side of the equation while maintaining the equality. The first step in this process is to eliminate the constant term on the left side of the equation, which is $-\frac{3}{2}$. To do this, we will add $\frac{3}{2}$ to both sides of the equation. This operation cancels out the $-\frac{3}{2}$ on the left side, moving us closer to isolating x. Adding $\frac{3}{2}$ to both sides gives us: $-\frac{2}{3}x - \frac{3}{2} + \frac{3}{2} = -\frac{6}{7} + \frac{3}{2}$. Simplifying this, we have $-\frac{2}{3}x = -\frac{6}{7} + \frac{3}{2}$. Next, we need to combine the fractions on the right side of the equation. To do this, we find a common denominator for 7 and 2, which is 14. We then convert the fractions to equivalent fractions with the common denominator: $-\frac{6}{7} = -\frac{12}{14}$ and $\frac{3}{2} = \frac{21}{14}$. Substituting these values back into the equation, we get $-\frac{2}{3}x = -\frac{12}{14} + \frac{21}{14}$. Now, we can add the fractions on the right side: $-\frac{2}{3}x = \frac{9}{14}$. We are now one step closer to solving for x. To isolate x completely, we need to eliminate the coefficient $-\frac{2}{3}$. To do this, we will multiply both sides of the equation by the reciprocal of $-\frac{2}{3}$, which is $-\frac{3}{2}$. Multiplying both sides by $-\frac{3}{2}$ gives us: $(-\frac{3}{2})(-\frac{2}{3}x) = (-\frac{3}{2})(\frac{9}{14})$. Simplifying the left side, we get $x$. On the right side, we multiply the fractions: $x = -\frac{27}{28}$. Thus, we have successfully isolated x and found its value. The solution to the equation is $x = -\frac{27}{28}$. This step-by-step process demonstrates the systematic approach to solving for x, highlighting the importance of each operation in isolating the variable.

Verification of the Solution

After solving for x and obtaining a potential solution, it's paramount to verify this solution. Verification is the crucial step that ensures the accuracy of our calculations and confirms that the value we found for x indeed satisfies the original equation. This process involves substituting the calculated value of x back into the original equation and checking if both sides of the equation are equal. If the equation holds true after the substitution, then our solution is correct. However, if the two sides are not equal, it indicates that an error has occurred during the solving process, and we need to revisit our steps to identify and correct the mistake. Let's apply this verification process to our solution, $x = -\frac{27}{28}$. We will substitute this value into the original equation: $-\frac{2}{3}x - \frac{3}{2} = -\frac{6}{7}$. Replacing x with $-\frac{27}{28}$, we get: $-\frac{2}{3}(-\frac{27}{28}) - \frac{3}{2} = -\frac{6}{7}$. Now, we simplify the left side of the equation. First, we multiply the fractions: $-\frac{2}{3}(-\frac{27}{28}) = \frac{54}{84}$. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 6: $\frac{54}{84} = \frac{9}{14}$. So, our equation now looks like this: $\frac{9}{14} - \frac{3}{2} = -\frac{6}{7}$. Next, we need to subtract $\frac{3}{2}$ from $\frac{9}{14}$. To do this, we find a common denominator for 14 and 2, which is 14. We convert the fractions to equivalent fractions with the common denominator: $\frac{3}{2} = \frac{21}{14}$. Substituting this back into the equation, we get: $\frac{9}{14} - \frac{21}{14} = -\frac{6}{7}$. Subtracting the fractions on the left side, we have: $-\frac{12}{14} = -\frac{6}{7}$. Finally, we simplify the fraction $-\frac{12}{14}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 2: $-\frac{12}{14} = -\frac{6}{7}$. Now, we can see that the left side of the equation, $-\frac{6}{7}$, is equal to the right side of the equation, $-\frac{6}{7}$. This confirms that our solution, $x = -\frac{27}{28}$, is correct. The verification process not only validates our solution but also reinforces our understanding of the equation-solving process. It provides a sense of closure and confidence in our answer, solidifying our grasp of algebraic techniques.

Common Mistakes and How to Avoid Them

When solving for x in equations, there are several common mistakes that students and even experienced mathematicians can make. Being aware of these pitfalls and understanding how to avoid them is crucial for accuracy and efficiency in problem-solving. One of the most frequent errors is incorrect application of the order of operations (PEMDAS/BODMAS). Forgetting to perform operations in the correct sequence can lead to significant errors in the final result. To avoid this, always double-check the order of operations before simplifying an expression or equation. Another common mistake is failing to perform the same operation on both sides of the equation. Remember, the golden rule of equation solving is to maintain balance by applying the same operation to both sides. Neglecting this principle will disrupt the equality and lead to an incorrect solution. It's also essential to be meticulous with signs, especially when dealing with negative numbers. A simple sign error can propagate through the entire solution, resulting in a wrong answer. To minimize sign errors, take your time, write out each step clearly, and double-check your work, paying close attention to the signs of each term. Mistakes in fraction arithmetic are also common, particularly when adding, subtracting, multiplying, or dividing fractions. Ensure that you find a common denominator before adding or subtracting fractions and that you correctly multiply or divide numerators and denominators. Practice and familiarity with fraction operations can significantly reduce these errors. Another pitfall is neglecting to distribute properly when dealing with parentheses or brackets. Remember to multiply each term inside the parentheses by the term outside. A failure to distribute correctly can lead to an incomplete or incorrect simplification of the equation. Finally, a crucial step that is often overlooked is the verification of the solution. Always substitute your solution back into the original equation to check for accuracy. This step can catch errors that you might have missed during the solving process. By being aware of these common mistakes and diligently applying the strategies to avoid them, you can enhance your accuracy and confidence in solving for x and tackling other algebraic problems. Consistent practice and attention to detail are key to mastering these skills and minimizing errors.

Conclusion

In conclusion, solving for x in the equation $-\frac{2}{3}x - \frac{3}{2} = -\frac{6}{7}$ is a process that requires a solid understanding of algebraic principles and careful application of equation-solving techniques. Throughout this guide, we have meticulously dissected each step, from understanding the basics of equation solving to verifying the final solution. We began by emphasizing the fundamental concept of maintaining balance in an equation, ensuring that any operation performed on one side is mirrored on the other. This principle is the cornerstone of all algebraic manipulations and is crucial for arriving at the correct solution. We then walked through the step-by-step process of isolating x, starting with the elimination of the constant term, $-\frac{3}{2}$, by adding its inverse to both sides. This was followed by combining fractions, multiplying by the reciprocal of the coefficient of x, and finally arriving at the solution: $x = -\frac{27}{28}$. The importance of verification was highlighted as the ultimate safeguard against errors. By substituting the calculated value of x back into the original equation, we confirmed that our solution indeed satisfied the equation, thereby validating our work. We also addressed common mistakes that often occur during equation solving, such as errors in the order of operations, sign errors, and incorrect fraction arithmetic. By being aware of these pitfalls and adopting strategies to avoid them, one can significantly improve accuracy and efficiency in solving algebraic problems. Mastering the skill of solving for x is not just about finding the solution to a specific problem; it's about developing a systematic approach to problem-solving that can be applied to a wide range of mathematical challenges. The principles and techniques discussed in this guide are applicable to various types of equations and algebraic problems, making this a valuable skill to acquire. As you continue your mathematical journey, remember that practice is the key to mastery. The more you practice solving for x and other algebraic problems, the more confident and proficient you will become. Embrace the challenges, learn from your mistakes, and celebrate your successes. With dedication and perseverance, you can unlock the power of algebra and excel in your mathematical pursuits.