Solving For X In The Equation 2(x+1) + 4x = 38
At the heart of algebra lies the ability to solve equations, a fundamental skill that unlocks doors to more advanced mathematical concepts. Solving equations involves finding the value(s) of the unknown variable(s) that make the equation true. In this comprehensive guide, we will delve into the step-by-step process of solving the equation 2(x+1) + 4x = 38, equipping you with the knowledge and confidence to tackle similar algebraic challenges. This equation, a linear equation in one variable, serves as an excellent example to illustrate the core principles of algebraic manipulation. We will break down each step with detailed explanations, ensuring a clear understanding of the underlying concepts. Whether you're a student learning algebra for the first time or someone seeking to refresh your skills, this guide offers a clear and concise approach to equation solving. By the end of this exploration, you'll not only be able to solve this specific equation but also grasp the broader strategies applicable to a wide range of algebraic problems. So, let's embark on this mathematical journey and unravel the solution to 2(x+1) + 4x = 38, empowering you with a solid foundation in algebra. Remember, the key to mastering algebra is practice and understanding the logic behind each step. With consistent effort and a clear understanding of the principles, you'll find yourself confidently solving even the most challenging equations.
Understanding the Equation
Before we jump into the solution, let's take a moment to understand the structure of the equation 2(x+1) + 4x = 38. This equation is a linear equation because the highest power of the variable 'x' is 1. It involves a combination of arithmetic operations, including multiplication, addition, and the distributive property. The equation presents a balance, with the left-hand side (2(x+1) + 4x) equaling the right-hand side (38). Our goal is to isolate 'x' on one side of the equation to determine its value. Understanding the equation is crucial as it allows us to strategize our approach. We need to carefully apply algebraic principles to simplify the equation while maintaining its balance. This involves using inverse operations to undo the operations that are performed on 'x'. For example, if 'x' is being multiplied by a number, we would divide both sides of the equation by that number. Similarly, if a number is being added to 'x', we would subtract that number from both sides. The distributive property, which we will use in the first step, is a key concept in simplifying algebraic expressions. It allows us to multiply a factor across a sum or difference within parentheses. By understanding these basic principles, we can approach the equation with a clear plan, ensuring that each step we take is logically sound and brings us closer to the solution. This initial assessment sets the stage for a systematic and successful solution process.
Step-by-Step Solution
Now, let's dive into the step-by-step solution of the equation 2(x+1) + 4x = 38. Each step is carefully explained to ensure clarity and understanding.
Step 1: Apply the Distributive Property
The first step in solving this equation is to simplify the left-hand side by applying the distributive property. The distributive property states that a(b + c) = ab + ac. In our equation, we have 2(x+1), so we need to distribute the 2 across the parentheses:
2(x + 1) = 2 * x + 2 * 1 = 2x + 2
Now, substitute this back into the original equation:
2x + 2 + 4x = 38
Applying the distributive property is a crucial first step as it eliminates the parentheses, making the equation easier to work with. This step allows us to combine like terms in the next stage, simplifying the equation further. The distributive property is a fundamental concept in algebra and is used extensively in simplifying expressions and solving equations. It's important to understand how it works and when to apply it. By correctly applying the distributive property, we've transformed the equation into a more manageable form, setting the stage for the next steps in the solution process. This initial simplification is key to efficiently solving the equation and avoiding potential errors.
Step 2: Combine Like Terms
The next step is to combine like terms on the left-hand side of the equation. Like terms are terms that have the same variable raised to the same power. In our equation, 2x and 4x are like terms. We can combine them by adding their coefficients:
2x + 4x = 6x
Now, substitute this back into the equation:
6x + 2 = 38
Combining like terms simplifies the equation by reducing the number of terms, making it easier to isolate the variable. This step is essential for streamlining the solution process and preventing unnecessary complexity. By combining like terms, we've effectively condensed the equation, bringing us closer to isolating 'x'. This process involves identifying terms with the same variable and exponent and then performing the indicated operation on their coefficients. The result is a more concise equation that is easier to manipulate. Understanding how to combine like terms is a fundamental skill in algebra and is crucial for solving a wide range of equations. By mastering this step, you can significantly simplify complex expressions and equations, making them more manageable and less prone to errors.
Step 3: Isolate the Variable Term
Our goal is to isolate 'x' on one side of the equation. To do this, we need to get rid of the constant term (+2) on the left-hand side. We can do this by subtracting 2 from both sides of the equation:
6x + 2 - 2 = 38 - 2
This simplifies to:
6x = 36
Isolating the variable term is a critical step in solving for 'x'. By subtracting 2 from both sides, we maintain the balance of the equation while effectively moving the constant term to the right-hand side. This process involves using inverse operations to undo the operations that are performed on the variable. In this case, we used subtraction to undo addition. Maintaining the balance of the equation is paramount in algebra, and performing the same operation on both sides ensures that the equation remains true. This step brings us one step closer to isolating 'x' and determining its value. The ability to isolate the variable term is a fundamental skill in algebra and is essential for solving a wide variety of equations. By mastering this technique, you can confidently manipulate equations to achieve your desired outcome.
Step 4: Solve for x
Finally, to solve for 'x', we need to isolate 'x' completely. We have 6x = 36. Since 'x' is being multiplied by 6, we can undo this by dividing both sides of the equation by 6:
6x / 6 = 36 / 6
This simplifies to:
x = 6
Therefore, the solution to the equation 2(x+1) + 4x = 38 is x = 6.
Solving for x involves performing the final step of isolating the variable by using the inverse operation of multiplication, which is division. By dividing both sides of the equation by 6, we effectively isolate 'x' and determine its value. This step represents the culmination of the algebraic manipulations we've performed, leading us to the solution. The solution, x = 6, is the value that, when substituted back into the original equation, will make the equation true. This final step underscores the importance of understanding inverse operations in algebra and their role in solving equations. By mastering this technique, you can confidently find the solution to a wide range of algebraic problems. The ability to solve for 'x' is a fundamental skill in mathematics and is essential for various applications in science, engineering, and other fields.
Verification
To ensure our solution is correct, we can substitute x = 6 back into the original equation and check if it holds true:
2(x + 1) + 4x = 38
2(6 + 1) + 4(6) = 38
2(7) + 24 = 38
14 + 24 = 38
38 = 38
The equation holds true, so our solution x = 6 is correct.
Verification is a crucial step in the problem-solving process, especially in mathematics. It involves substituting the solution we obtained back into the original equation to ensure that it satisfies the equation. In this case, we substituted x = 6 back into 2(x+1) + 4x = 38 and found that both sides of the equation equal 38. This confirms that our solution is correct and that we have successfully solved the equation. Verification not only provides confidence in our answer but also helps to catch any errors that may have occurred during the solution process. It's a valuable habit to develop in mathematics as it ensures accuracy and promotes a deeper understanding of the concepts involved. By verifying our solution, we solidify our understanding of the problem and the steps we took to solve it.
Conclusion
In this comprehensive guide, we have successfully solved the equation 2(x+1) + 4x = 38 using a step-by-step approach. We began by understanding the structure of the equation and then systematically applied algebraic principles to isolate the variable 'x'. We utilized the distributive property, combined like terms, and used inverse operations to arrive at the solution x = 6. Finally, we verified our solution by substituting it back into the original equation, confirming its correctness.
In conclusion, solving equations is a fundamental skill in algebra and mathematics as a whole. The process involves a series of logical steps, each building upon the previous one. By mastering these steps, you can confidently solve a wide range of equations and apply these skills to more complex mathematical problems. The key to success lies in understanding the underlying principles, practicing consistently, and developing a systematic approach to problem-solving. Remember, each equation presents a unique challenge, but with a solid foundation in algebra and a methodical approach, you can confidently tackle any algebraic problem that comes your way. This exercise in solving 2(x+1) + 4x = 38 serves as a valuable example of the problem-solving process in algebra and reinforces the importance of accuracy, attention to detail, and a clear understanding of algebraic principles.
