Solving (1/12)(7x+6) - 3 = (1/2)(-x+6) A Step-by-Step Solution

This article dives deep into the process of solving the linear equation (1/12)(7x + 6) - 3 = (1/2)(-x + 6), offering a detailed, step-by-step solution and clear explanations for each step. Linear equations are fundamental in mathematics and have wide applications in various fields, making it crucial to master the techniques for solving them. Our goal here is to provide a comprehensive guide that not only leads to the solution x = 66/13 but also enhances your understanding of the underlying mathematical principles. We'll break down the equation into manageable parts, explain the logic behind each operation, and highlight common pitfalls to avoid. Whether you're a student learning algebra or someone looking to refresh your math skills, this article will provide you with the knowledge and confidence to tackle similar problems.

Understanding the Equation's Structure

Before we jump into solving the equation, let's take a moment to understand its structure. We have a linear equation involving fractions, parentheses, and variables. The presence of fractions might seem intimidating at first, but we'll see how to deal with them effectively. The parentheses indicate the need for distribution, a crucial step in simplifying the equation. Our main objective is to isolate the variable 'x' on one side of the equation, which will reveal its value. To achieve this, we'll employ a combination of algebraic operations, ensuring that we maintain the equality throughout the process. Remember, whatever operation we perform on one side of the equation, we must also perform on the other side. This fundamental principle is the key to solving any equation. We will start by simplifying both sides of the equation, then combine like terms, and finally isolate 'x'. This methodical approach will ensure accuracy and clarity in our solution.

Step 1: Distributing and Simplifying the Equation

Our first step in solving the equation (1/12)(7x + 6) - 3 = (1/2)(-x + 6) involves distributing the fractions on both sides. This means multiplying the terms inside the parentheses by the fractions outside. On the left side, we distribute (1/12) to both 7x and 6, resulting in (7x/12) + (6/12). On the right side, we distribute (1/2) to both -x and 6, resulting in (-x/2) + 3. Now, our equation looks like this: (7x/12) + (6/12) - 3 = (-x/2) + 3. Notice that we still have fractions and a constant term (-3) on the left side. To further simplify, we can reduce the fraction 6/12 to 1/2. So, the equation becomes: (7x/12) + (1/2) - 3 = (-x/2) + 3. Next, we can combine the constant terms on the left side. We have 1/2 - 3, which is equivalent to 1/2 - 6/2, resulting in -5/2. Our equation is now: (7x/12) - (5/2) = (-x/2) + 3. This simplified form makes it easier to proceed with the next steps, which will involve eliminating the fractions and isolating the variable 'x'. Remember, the goal of this step is to reduce the complexity of the equation, making it more manageable and less prone to errors.

Step 2: Eliminating Fractions for Clarity

To eliminate the fractions in the equation (7x/12) - (5/2) = (-x/2) + 3, we need to find the least common multiple (LCM) of the denominators. The denominators in our equation are 12 and 2. The LCM of 12 and 2 is 12. This means we will multiply every term in the equation by 12. Multiplying each term by 12, we get: 12 * (7x/12) - 12 * (5/2) = 12 * (-x/2) + 12 * 3. Now, we can simplify each term: The first term, 12 * (7x/12), simplifies to 7x because the 12s cancel out. The second term, 12 * (5/2), simplifies to 30 because 12 divided by 2 is 6, and 6 times 5 is 30. The third term, 12 * (-x/2), simplifies to -6x because 12 divided by 2 is 6, and 6 times -x is -6x. The fourth term, 12 * 3, simplifies to 36. So, our equation now looks like this: 7x - 30 = -6x + 36. By multiplying by the LCM, we have successfully eliminated the fractions, making the equation much easier to work with. This step is crucial because it transforms a complex equation into a simpler one, reducing the likelihood of making mistakes in subsequent steps. We're now one step closer to isolating 'x' and finding its value.

Step 3: Isolating the Variable x

Now that we have the equation 7x - 30 = -6x + 36, our next goal is to isolate the variable 'x'. This means getting all the terms with 'x' on one side of the equation and all the constant terms on the other side. To do this, we can add 6x to both sides of the equation. This will eliminate the -6x term on the right side and move the 'x' term to the left side. Adding 6x to both sides gives us: 7x + 6x - 30 = -6x + 6x + 36. Simplifying this, we get: 13x - 30 = 36. Now, we need to move the constant term (-30) to the right side of the equation. To do this, we can add 30 to both sides: 13x - 30 + 30 = 36 + 30. Simplifying, we get: 13x = 66. We're almost there! We now have 13x equal to 66. To isolate 'x' completely, we need to divide both sides of the equation by 13. This will give us the value of 'x'. Dividing both sides by 13, we get: (13x)/13 = 66/13. Simplifying, we find: x = 66/13. We have successfully isolated 'x' and found its value. This step-by-step process of moving terms and simplifying the equation is fundamental to solving linear equations. By carefully performing each operation, we arrive at the solution with confidence.

Step 4: Verifying the Solution

After finding the solution x = 66/13, it's always a good practice to verify if our solution is correct. This involves substituting the value of x back into the original equation and checking if both sides of the equation are equal. Our original equation was: (1/12)(7x + 6) - 3 = (1/2)(-x + 6). Let's substitute x = 66/13 into the equation: (1/12)(7*(66/13) + 6) - 3 = (1/2)(-(66/13) + 6). Now, we need to simplify both sides of the equation. On the left side, we have: (1/12)((462/13) + 6) - 3. To add the terms inside the parentheses, we need a common denominator, which is 13. So, we rewrite 6 as 78/13: (1/12)((462/13) + (78/13)) - 3. Adding the fractions inside the parentheses, we get: (1/12)(540/13) - 3. Multiplying (1/12) by (540/13), we get: 540/156 - 3. Now, we simplify the fraction 540/156 by dividing both the numerator and the denominator by their greatest common divisor, which is 12. This gives us: 45/13 - 3. To subtract 3, we rewrite it as a fraction with a denominator of 13: 45/13 - 39/13. Subtracting the fractions, we get: 6/13. So, the left side of the equation simplifies to 6/13. Now, let's simplify the right side of the equation: (1/2)(-(66/13) + 6). Again, we need a common denominator to add the terms inside the parentheses. We rewrite 6 as 78/13: (1/2)(-(66/13) + (78/13)). Adding the fractions inside the parentheses, we get: (1/2)(12/13). Multiplying (1/2) by (12/13), we get: 12/26. Simplifying this fraction by dividing both the numerator and the denominator by 2, we get: 6/13. So, the right side of the equation also simplifies to 6/13. Since both sides of the equation are equal (6/13 = 6/13), our solution x = 66/13 is correct. Verifying the solution is a crucial step in problem-solving. It ensures that we haven't made any errors in our calculations and that our answer is indeed the correct solution to the equation.

Common Pitfalls and How to Avoid Them

Solving linear equations can sometimes be tricky, and there are common pitfalls that students often encounter. Being aware of these pitfalls and knowing how to avoid them can significantly improve your accuracy and problem-solving skills. One common mistake is incorrectly distributing. Remember to multiply the term outside the parentheses by every term inside the parentheses. For example, in the equation (1/12)(7x + 6), you need to multiply (1/12) by both 7x and 6. Another common pitfall is making errors when dealing with negative signs. Pay close attention to the signs of each term and ensure you are applying the correct operations. For instance, when moving a term from one side of the equation to the other, remember to change its sign. For example, if you have -30 on one side, it becomes +30 when moved to the other side. Failing to find a common denominator when adding or subtracting fractions is another frequent error. Before you can add or subtract fractions, they must have the same denominator. Make sure you find the least common multiple (LCM) of the denominators and convert the fractions accordingly. Another pitfall is not simplifying fractions completely. Always simplify fractions to their lowest terms to make calculations easier and to avoid errors. For example, 12/26 can be simplified to 6/13. Finally, a very important step is to forgetting to verify the solution. Always substitute your solution back into the original equation to check if it is correct. This will help you catch any mistakes you might have made during the solving process. By being mindful of these common pitfalls and taking steps to avoid them, you can become more confident and successful in solving linear equations. Remember, practice makes perfect, so the more you solve these equations, the better you'll become at it.

Conclusion: Mastering Linear Equations

In conclusion, solving the linear equation (1/12)(7x + 6) - 3 = (1/2)(-x + 6) to find x = 66/13 involves a series of methodical steps, each requiring careful attention to detail. We began by distributing fractions, which simplified the equation, setting the stage for further operations. We then eliminated the fractions by multiplying by the least common multiple, a crucial step in making the equation more manageable. This was followed by isolating the variable 'x' through strategic addition and division, revealing the solution. Finally, we verified our solution, a critical step to ensure accuracy and validate our calculations. The entire process demonstrates the power of algebraic manipulation and the importance of following a systematic approach. Linear equations are a cornerstone of mathematics, serving as building blocks for more complex concepts. Mastering the techniques to solve them not only enhances your mathematical abilities but also equips you with problem-solving skills applicable across various disciplines. The common pitfalls we discussed, such as incorrect distribution, sign errors, and failure to find common denominators, highlight the importance of careful execution and attention to detail. Verifying the solution reinforces the need for thoroughness and accuracy. By understanding these challenges and practicing regularly, you can develop confidence in solving linear equations and tackle more advanced mathematical problems. This journey through solving a linear equation is more than just finding an answer; it's about cultivating a mindset of precision, logical thinking, and perseverance—qualities that are valuable in all aspects of life. So, embrace the challenges, practice diligently, and celebrate your successes as you continue to explore the fascinating world of mathematics.