Solving 4(3x-6) = 24 A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill. It's like deciphering a secret code, where you need to isolate a variable to uncover its value. This article breaks down the steps involved in solving the equation $4(3x - 6) = 24$, providing a clear and comprehensive guide for anyone looking to enhance their equation-solving abilities. We'll delve into each step, explaining the reasoning behind it, so you not only understand the how but also the why. Let's embark on this mathematical journey together!

Original Equation: 4(3x - 6) = 24

The journey of solving any equation begins with the original equation. In our case, it's $4(3x - 6) = 24$. This equation presents a scenario where a variable, x, is entangled in a series of operations. Our goal is to isolate x on one side of the equation to determine its value. Before we dive into the steps, it's crucial to understand the basic principles that govern equation solving. These principles are rooted in the concept of maintaining balance. Think of an equation as a weighing scale; to keep it balanced, any operation performed on one side must also be performed on the other. This fundamental concept underpins every step we'll take in solving this equation. We will need to apply the distributive property, combine like terms, and use inverse operations to ultimately find the value of x that satisfies the equation. Understanding the structure of the equation is the first key to unlocking its solution. Remember, each term plays a vital role, and by carefully manipulating these terms, we can reveal the hidden value of x.

Step 1: 12x - 24 = 24 (Applying the Distributive Property)

In this crucial Step 1, we employ the distributive property. This property allows us to eliminate the parentheses in the equation, making it easier to work with. The distributive property states that $a(b + c) = ab + ac$. Applying this to our equation, $4(3x - 6) = 24$, we multiply the 4 by both terms inside the parentheses: 4 multiplied by 3x gives us 12x, and 4 multiplied by -6 gives us -24. This transforms the equation into $12x - 24 = 24$. Understanding the distributive property is essential for simplifying algebraic expressions and equations. It allows us to break down complex expressions into simpler, more manageable terms. This step is a significant move towards isolating x, as it removes the initial barrier of the parentheses. By correctly applying the distributive property, we pave the way for the subsequent steps in solving the equation. It's a fundamental algebraic technique that's used extensively in various mathematical contexts. This step demonstrates how a seemingly complex equation can be simplified by applying basic algebraic principles. Remember, the goal is to unravel the equation systematically, and this step takes us closer to that goal.

Step 2: 12x - 24 + 24 = 24 + 24 (Isolating the Variable Term)

Step 2 is a crucial step in isolating the term containing our variable, x. We aim to get the term $12x$ by itself on one side of the equation. To achieve this, we employ the concept of inverse operations. Since we have $12x - 24$, we need to eliminate the -24. The inverse operation of subtraction is addition, so we add 24 to both sides of the equation. This gives us $12x - 24 + 24 = 24 + 24$. Adding 24 to both sides maintains the balance of the equation, a fundamental principle in solving equations. The -24 and +24 on the left side cancel each other out, leaving us with just $12x$. On the right side, $24 + 24$ equals 48. Therefore, after this step, our equation simplifies to $12x = 48$. This step highlights the importance of using inverse operations to isolate the variable term. By strategically adding 24 to both sides, we've successfully removed the constant term from the left side, bringing us closer to solving for x. This step is a testament to the power of algebraic manipulation in simplifying equations.

Step 3: 12x = 48 (Simplified Equation)

In Step 3, we have successfully simplified the equation to $12x = 48$. This is a significant milestone in our equation-solving journey. We've managed to isolate the term containing x on one side of the equation, which brings us closer to finding the value of x itself. The equation $12x = 48$ now presents a clear relationship: 12 times x equals 48. To find the value of x, we need to undo the multiplication by 12. This step demonstrates the power of simplification in equation solving. By systematically applying algebraic principles, we've transformed the original equation into a much simpler form. This simplified equation makes it easier to see the next step required to solve for x. It's a clear illustration of how each step in the equation-solving process builds upon the previous one, leading us closer to the solution. The equation $12x = 48$ is a stepping stone to the final answer, and we're now well-positioned to determine the value of x.

Step 4: 12x / 12 = 48 / 12 (Dividing to Isolate x)

Step 4 is the final step in isolating x and finding its value. We have the equation $12x = 48$, which means 12 multiplied by x equals 48. To isolate x, we need to perform the inverse operation of multiplication, which is division. We divide both sides of the equation by 12. This gives us $rac{12x}{12} = rac{48}{12}$. Dividing both sides by the same number maintains the balance of the equation, a fundamental principle in algebra. On the left side, the 12 in the numerator and the 12 in the denominator cancel each other out, leaving us with just x. On the right side, 48 divided by 12 equals 4. Therefore, the equation simplifies to $x = 4$. This step demonstrates the power of using inverse operations to solve for a variable. By dividing both sides by 12, we've successfully isolated x and found its value. This final step is the culmination of all the previous steps, showcasing the systematic approach to solving equations. The solution, $x = 4$, is the value that satisfies the original equation, making the equation a true statement.

Solution: x = 4

We've reached the culmination of our mathematical journey: the solution. After meticulously following each step, we've determined that $x = 4$. This means that 4 is the value that, when substituted back into the original equation, $4(3x - 6) = 24$, makes the equation a true statement. To verify our solution, we can substitute $x = 4$ back into the original equation: $4(3(4) - 6) = 24$ simplifies to $4(12 - 6) = 24$, which further simplifies to $4(6) = 24$, and finally, $24 = 24$. This confirms that our solution is correct. The process of solving equations is a fundamental skill in mathematics, and this example illustrates the systematic approach required to find the solution. Each step, from applying the distributive property to using inverse operations, plays a crucial role in isolating the variable. The solution, $x = 4$, is not just a number; it's the answer to a mathematical puzzle, a testament to the power of algebraic manipulation. Understanding and mastering these steps is essential for success in algebra and beyond. Congratulations on solving this equation!

Conclusion: Mastering Equation-Solving Techniques

In conclusion, solving the equation $4(3x - 6) = 24$ is a journey through the core principles of algebra. We started with the original equation and, through a series of logical steps, arrived at the solution, $x = 4$. This process highlights the importance of understanding and applying fundamental concepts such as the distributive property, inverse operations, and maintaining the balance of the equation. Each step we took was deliberate and purposeful, building upon the previous one to gradually isolate the variable x. The distributive property allowed us to simplify the equation by removing the parentheses. Adding 24 to both sides helped us isolate the term containing x. Dividing both sides by 12 was the final step in determining the value of x. This step-by-step approach is not just applicable to this specific equation; it's a framework for solving a wide range of algebraic equations. Mastering these techniques is crucial for success in mathematics, as equation solving is a fundamental skill that underpins many other mathematical concepts. By understanding the underlying principles and practicing regularly, anyone can become proficient in solving equations. Remember, mathematics is not just about finding the right answer; it's about understanding the process and the reasoning behind it. The journey of solving $4(3x - 6) = 24$ is a perfect illustration of this principle. Keep practicing, keep exploring, and keep solving!