Solving The Equation 6(x-5) = 4x + 20 A Step-by-Step Guide
In this article, we will delve into the process of solving the equation 6(x-5) = 4x + 20. This equation falls under the category of linear equations, which are fundamental in algebra and have widespread applications in various fields such as physics, engineering, economics, and computer science. Understanding how to solve linear equations is a crucial skill for anyone pursuing studies or careers in these areas. This problem involves a single variable, x, and our goal is to isolate x on one side of the equation to find its value. By the end of this guide, you will not only know the solution but also understand the underlying principles and steps involved in solving similar equations. We will break down the equation step by step, ensuring a clear and comprehensive understanding of each operation performed. The techniques we use here can be applied to a multitude of other algebraic problems, making this a valuable learning experience. Let's begin by examining the equation closely and identifying the initial steps required to simplify it. Remember, solving equations is like piecing together a puzzle, and each step brings us closer to the final solution. The beauty of algebra lies in its systematic approach, where following the correct steps leads us to the correct answer. So, let's embark on this algebraic journey together and demystify the process of solving this equation.
Step 1: Distribute the 6 on the Left Side
Our first step in solving the equation 6(x-5) = 4x + 20 is to simplify the left side by distributing the 6 across the terms inside the parentheses. This involves multiplying 6 by both x and -5. The distributive property states that a(b + c) = ab + ac. Applying this property here, we multiply 6 by x, which gives us 6x, and then we multiply 6 by -5, which results in -30. So, the left side of the equation becomes 6x - 30. Now our equation looks like this: 6x - 30 = 4x + 20. This step is crucial because it removes the parentheses, making it easier to combine like terms later on. Distributing correctly ensures that we maintain the equality of the equation, as we are performing the same operation on both terms inside the parentheses. This is a fundamental technique in algebra, and mastering it is essential for solving more complex equations. By expanding the expression on the left side, we've effectively transformed the equation into a more manageable form. The next step will involve gathering the x terms on one side and the constant terms on the other. This process of distribution is not just a mechanical step; it's about understanding how multiplication interacts with addition and subtraction within an algebraic expression. It's a foundational skill that paves the way for further algebraic manipulations.
Step 2: Move the 4x Term to the Left Side
Having simplified the left side of the equation to 6x - 30 = 4x + 20, our next goal is to consolidate the terms containing x on one side of the equation. Conventionally, we aim to move the x terms to the left side. To achieve this, we need to eliminate the 4x term from the right side. We can do this by subtracting 4x from both sides of the equation. This is based on the fundamental principle of algebraic manipulation: whatever operation we perform on one side of the equation, we must also perform on the other side to maintain the equality. Subtracting 4x from both sides gives us: 6x - 30 - 4x = 4x + 20 - 4x. On the left side, we have 6x - 4x, which simplifies to 2x. On the right side, 4x - 4x cancels out, leaving us with just 20. So, our equation now looks like this: 2x - 30 = 20. This step is crucial because it groups the variable terms together, making it easier to isolate x later on. By subtracting 4x from both sides, we have effectively moved the x term from the right to the left side, simplifying the equation and bringing us closer to the solution. This process of rearranging terms is a key technique in solving algebraic equations, allowing us to manipulate the equation into a form where the variable can be easily isolated. Understanding this step is essential for mastering the art of equation solving.
Step 3: Move the Constant Term to the Right Side
Now that we have the equation in the form 2x - 30 = 20, our next task is to isolate the term with x on the left side. To do this, we need to move the constant term, -30, to the right side of the equation. We can accomplish this by adding 30 to both sides of the equation. Again, this maintains the balance of the equation, as we are performing the same operation on both sides. Adding 30 to both sides gives us: 2x - 30 + 30 = 20 + 30. On the left side, -30 + 30 cancels out, leaving us with just 2x. On the right side, 20 + 30 equals 50. So, our equation now simplifies to: 2x = 50. This step is a critical one in the process of solving for x. By moving the constant term to the right side, we have successfully isolated the term containing x on the left side. This makes the final step of dividing by the coefficient of x much more straightforward. The principle of adding the same value to both sides is a fundamental technique in algebra, ensuring that the equation remains balanced throughout the manipulation. This step demonstrates the power of inverse operations in solving equations, as we use addition to undo the subtraction of 30. Understanding this concept is vital for tackling more complex algebraic problems.
Step 4: Divide Both Sides by the Coefficient of x
With the equation now simplified to 2x = 50, the final step in solving for x is to isolate x completely. Currently, x is being multiplied by 2. To undo this multiplication, we need to perform the inverse operation, which is division. We divide both sides of the equation by 2, the coefficient of x. Dividing both sides by 2 gives us: (2x) / 2 = 50 / 2. On the left side, the 2 in the numerator and the 2 in the denominator cancel each other out, leaving us with just x. On the right side, 50 divided by 2 equals 25. Therefore, the solution to the equation is x = 25. This final step highlights the importance of inverse operations in solving equations. By dividing both sides by the coefficient of x, we have successfully isolated x and found its value. This is a fundamental technique in algebra and is applicable to a wide range of equations. The solution x = 25 means that if we substitute 25 for x in the original equation, both sides of the equation will be equal. This can be verified by plugging 25 back into the original equation: 6(25 - 5) = 4(25) + 20. Simplifying this, we get 6(20) = 100 + 20, which further simplifies to 120 = 120, confirming that our solution is correct. This process of verification is a good practice to ensure the accuracy of the solution.
After meticulously working through each step, we have arrived at the solution to the equation 6(x-5) = 4x + 20. By distributing, rearranging terms, and using inverse operations, we successfully isolated x and found its value. The solution is x = 25. This means that the value of x that satisfies the given equation is 25. We can confidently state that option A, x = 25, is the correct answer. Understanding the process of solving linear equations is not just about finding the right answer; it's about developing a systematic approach to problem-solving. Each step we took, from distributing the 6 to dividing by the coefficient of x, was a deliberate action aimed at simplifying the equation and isolating the variable. This step-by-step methodology is crucial for tackling more complex algebraic problems in the future. Moreover, the ability to solve equations is a foundational skill that extends beyond mathematics. It is applicable in various fields, including science, engineering, economics, and computer science, where mathematical models are used to represent and solve real-world problems. Therefore, mastering the techniques discussed in this article will not only help you succeed in algebra but also equip you with valuable problem-solving skills that are transferable to other domains.
In conclusion, we have successfully navigated the process of solving the linear equation 6(x-5) = 4x + 20. We began by understanding the importance of solving such equations and their relevance in various fields. Each step was carefully explained, starting with distributing the 6, moving the x terms to one side, moving the constant terms to the other side, and finally, dividing by the coefficient of x. Through these steps, we methodically isolated x and determined its value to be 25. This solution not only answers the specific question but also reinforces the fundamental principles of algebraic manipulation. The distributive property, combining like terms, and the use of inverse operations are all essential techniques that are widely applicable in mathematics. Moreover, the systematic approach we adopted, breaking down the problem into smaller, manageable steps, is a valuable problem-solving strategy that can be applied in various contexts. By understanding the underlying logic and reasoning behind each step, you can confidently tackle similar equations and even more complex problems in algebra. The journey of solving this equation is a testament to the power of mathematical principles and the importance of a structured approach. As you continue your mathematical journey, remember that each equation is a puzzle waiting to be solved, and with the right tools and techniques, you can unlock its solution.
