Solving For X In The Equation (1/3)(12x-24)=16

In this article, we will walk through the process of solving for the value of x in the equation $\frac{1}{3}(12x - 24) = 16$. This is a common type of algebraic problem that involves simplifying expressions and isolating the variable. Understanding how to solve such equations is crucial for various mathematical and real-world applications. We will break down each step, providing a clear and concise explanation to help you grasp the underlying concepts. By the end of this article, you should be well-equipped to tackle similar problems with confidence. Solving algebraic equations is a fundamental skill in mathematics, and mastering it opens doors to more advanced topics. So, let's dive in and unravel this equation step by step.

Understanding the Equation

Before we begin solving, let's take a closer look at the equation: $\frac{1}{3}(12x - 24) = 16$. This equation involves a variable x, which is what we need to find. The equation states that one-third of the expression 12x - 24 is equal to 16. Our goal is to isolate x on one side of the equation to determine its value. To do this, we will use algebraic principles such as the distributive property and inverse operations. The distributive property allows us to multiply a number by a group of terms, while inverse operations help us to undo mathematical operations. For example, to undo multiplication, we use division, and to undo subtraction, we use addition. By applying these principles systematically, we can simplify the equation and solve for x. It is important to maintain balance in the equation, meaning that any operation performed on one side must also be performed on the other side. This ensures that the equality remains valid throughout the solving process. Understanding the structure of the equation and the principles we will use is the first step towards finding the solution.

Step-by-Step Solution

Step 1: Distribute the 1/3

The first step in solving the equation $\frac{1}{3}(12x - 24) = 16$ is to distribute the $\frac{1}{3}$ across the terms inside the parentheses. This means we multiply both 12x and -24 by $\frac{1}{3}$.

• $\frac{1}{3} * 12x = 4x$
• $\frac{1}{3} * -24 = -8$

So, after distributing, the equation becomes:

$4x - 8 = 16$

Distributing the $\frac{1}{3}$ simplifies the equation by removing the parentheses, making it easier to isolate the variable x. This step is crucial because it transforms the equation into a more manageable form. By applying the distributive property, we ensure that each term within the parentheses is correctly accounted for, maintaining the balance of the equation. This foundational step sets the stage for the subsequent steps in solving for x.

Step 2: Isolate the Term with x

Now that we have the equation $4x - 8 = 16$, our next step is to isolate the term containing x, which is 4x. To do this, we need to eliminate the -8 on the left side of the equation. We can accomplish this by adding 8 to both sides of the equation. Adding the same value to both sides maintains the equality and helps us move closer to isolating x.

$4x - 8 + 8 = 16 + 8$

This simplifies to:

$4x = 24$

By adding 8 to both sides, we have successfully isolated the term 4x on the left side of the equation. This step is a key part of the solving process, as it brings us closer to finding the value of x. Isolating the variable term is a common strategy in algebra, and it is essential for solving various types of equations. With 4x now isolated, we can proceed to the final step of dividing to find the value of x.

Step 3: Solve for x

We have now arrived at the equation $4x = 24$. To solve for x, we need to isolate x completely. Since x is being multiplied by 4, we can undo this operation by dividing both sides of the equation by 4. Dividing both sides by the same number keeps the equation balanced and allows us to find the value of x.

$\frac{4x}{4} = \frac{24}{4}$

This simplifies to:

$x = 6$

Therefore, the value of x that satisfies the equation is 6. This final step completes the solution, giving us the numerical value of x. By dividing both sides by 4, we have effectively isolated x, revealing its value. This solution is the answer to the original problem and demonstrates the power of algebraic manipulation in solving equations.

Final Answer

After carefully following the steps, we have determined that the value of x in the equation $\frac{1}{3}(12x - 24) = 16$ is 6. This means that when x is replaced with 6 in the original equation, the equation holds true. To verify our solution, we can substitute x = 6 back into the original equation and check if both sides are equal.

$\frac{1}{3}(12(6) - 24) = 16$

$\frac{1}{3}(72 - 24) = 16$

$\frac{1}{3}(48) = 16$

$16 = 16$

Since the equation holds true, our solution x = 6 is correct. This verification step is an important practice to ensure the accuracy of our solution. In conclusion, by distributing, isolating terms, and using inverse operations, we have successfully solved the equation and found the value of x.

Conclusion

In this article, we have demonstrated how to solve for x in the equation $\frac{1}{3}(12x - 24) = 16$. By applying algebraic principles such as the distributive property and inverse operations, we systematically simplified the equation and isolated the variable. The steps included distributing $\frac{1}{3}$ across the terms in the parentheses, isolating the term with x by adding 8 to both sides, and finally, solving for x by dividing both sides by 4. This process led us to the solution x = 6, which we verified by substituting back into the original equation. Understanding these steps and the underlying principles is crucial for solving various algebraic equations. Mastering these techniques not only helps in academic settings but also in real-world problem-solving scenarios. By practicing and applying these methods, you can build confidence in your algebraic skills and tackle more complex problems effectively. This problem serves as a good example of how algebraic equations can be solved with a clear and methodical approach. Keep practicing, and you'll become more proficient in solving such equations.