Solving For X In (2)(6) + (3)(x) = 0 A Step-by-Step Guide

In this article, we will delve into the mathematical problem presented by the equation $(2)(6) + (3)(x) = 0$. Our primary goal is to find the value of $x$ that satisfies this equation. This is a fundamental algebra problem that requires a clear understanding of basic arithmetic operations and algebraic manipulation. We will break down the problem step by step, ensuring that each step is clearly explained and justified. This approach will not only help you understand the solution to this specific problem but also equip you with the skills to tackle similar algebraic equations. Let's embark on this mathematical journey to unlock the value of $x$.

Understanding the Equation: A Foundation for Success

The equation we are presented with, $(2)(6) + (3)(x) = 0$, is a linear equation. Linear equations are algebraic equations where the highest power of the variable (in this case, $x$) is 1. These equations are fundamental in mathematics and have wide-ranging applications in various fields, including physics, engineering, and economics. To solve for $x$, we need to isolate it on one side of the equation. This involves performing operations on both sides of the equation to maintain the balance. The key principle here is that whatever operation you perform on one side of the equation, you must perform the same operation on the other side. This ensures that the equality remains valid. Before we dive into the algebraic manipulations, let's first simplify the equation by performing the multiplication operations that are immediately apparent. This will make the equation easier to work with and bring us closer to isolating the variable $x$.

Simplifying the Equation: The First Step Towards Isolation

The first step in solving the equation $(2)(6) + (3)(x) = 0$ is to simplify the multiplication. We start by multiplying 2 and 6, which gives us 12. So, the equation now becomes $12 + 3x = 0$. This simplification is crucial because it reduces the complexity of the equation and makes it easier to isolate the term containing $x$. By performing this basic arithmetic operation, we have effectively taken the first step towards solving for $x$. The equation $12 + 3x = 0$ is now in a form that allows us to proceed with algebraic manipulations to isolate $x$. Remember, the goal is to get $x$ by itself on one side of the equation. This involves using inverse operations, which we will explore in the next section. Understanding this initial simplification is vital for grasping the subsequent steps in the solution process. Let's move forward and see how we can further isolate $x$ and find its value.

Isolating the Term with x: Applying Inverse Operations

Now that we have the simplified equation $12 + 3x = 0$, our next goal is to isolate the term containing $x$, which is $3x$. To do this, we need to eliminate the constant term, 12, from the left side of the equation. We can achieve this by using the inverse operation of addition, which is subtraction. We subtract 12 from both sides of the equation. This maintains the balance of the equation and moves us closer to isolating $x$. Subtracting 12 from both sides gives us: $12 + 3x - 12 = 0 - 12$. This simplifies to $3x = -12$. We have now successfully isolated the term with $x$ on one side of the equation. The next step is to isolate $x$ itself. This involves dealing with the coefficient of $x$, which is 3. Understanding the concept of inverse operations is crucial in algebra, and this step demonstrates its practical application in solving equations. Let's proceed to the final step of isolating $x$ by dealing with its coefficient.

Solving for x: The Final Calculation

We've arrived at the equation $3x = -12$. To finally solve for $x$, we need to isolate $x$ completely. This means we need to get rid of the coefficient 3 that is multiplying $x$. The inverse operation of multiplication is division. Therefore, we will divide both sides of the equation by 3. This maintains the balance of the equation and isolates $x$. Dividing both sides by 3 gives us: $\frac{3x}{3} = \frac{-12}{3}$. This simplifies to $x = -4$. We have now successfully solved for $x$. The value of $x$ that satisfies the original equation $(2)(6) + (3)(x) = 0$ is -4. This final calculation demonstrates the power of algebraic manipulation and the importance of using inverse operations to isolate the variable we are trying to find. Let's summarize the steps we took to arrive at this solution and reinforce our understanding of the process.

Summary of Steps: A Recap of the Solution Process

Let's recap the steps we took to solve the equation $(2)(6) + (3)(x) = 0$: 1. Simplify the equation: We started by performing the multiplication $(2)(6)$ to get 12, resulting in the equation $12 + 3x = 0$. 2. Isolate the term with x: We subtracted 12 from both sides of the equation to isolate the term $3x$, resulting in $3x = -12$. 3. Solve for x: We divided both sides of the equation by 3 to isolate $x$, resulting in $x = -4$. By following these steps, we successfully found the value of $x$ that satisfies the equation. This process highlights the systematic approach required to solve algebraic equations. Each step builds upon the previous one, leading us closer to the solution. Understanding this step-by-step approach is crucial for tackling more complex algebraic problems. Now, let's verify our solution to ensure its accuracy.

Verification: Ensuring Accuracy and Confidence

To ensure the accuracy of our solution, it's essential to verify that the value we found for $x$, which is -4, indeed satisfies the original equation. This involves substituting $x = -4$ back into the equation $(2)(6) + (3)(x) = 0$ and checking if the equation holds true. Substituting $x = -4$ gives us: $(2)(6) + (3)(-4) = 0$. Simplifying this, we get: $12 + (-12) = 0$, which further simplifies to $0 = 0$. This confirms that our solution, $x = -4$, is correct. Verification is a crucial step in problem-solving as it provides confidence in the accuracy of the solution. It also helps in identifying any potential errors in the solution process. By verifying our solution, we can be certain that we have correctly solved the equation. In conclusion, we have successfully solved for $x$ in the equation $(2)(6) + (3)(x) = 0$, and we have verified our solution to ensure its accuracy. This problem demonstrates the fundamental principles of algebra and the importance of a systematic approach to problem-solving.

Conclusion: Mastering Algebraic Equations

In this comprehensive guide, we have successfully navigated the process of solving the equation $(2)(6) + (3)(x) = 0$. We began by simplifying the equation, then systematically isolated the term containing $x$, and finally solved for $x$. We also emphasized the importance of verifying the solution to ensure accuracy. The solution we found is $x = -4$. This exercise has highlighted the fundamental principles of algebra, including the use of inverse operations and the importance of maintaining balance in an equation. By mastering these principles, you can confidently tackle a wide range of algebraic problems. Remember, practice is key to developing your algebraic skills. The more you practice, the more comfortable and proficient you will become in solving equations. We encourage you to apply these techniques to other similar problems and continue to build your mathematical foundation. This journey into solving algebraic equations is just the beginning. There are many more exciting mathematical concepts to explore and master. Keep learning, keep practicing, and keep pushing your mathematical boundaries.