Solve The Expression 2x + 3(x - 11) When X = 6.
In this article, we will delve into the step-by-step solution of the algebraic expression 2x + 3(x - 11) when the value of x is given as 6. This type of problem is a fundamental concept in algebra, and understanding how to solve it is crucial for mastering more complex algebraic equations. We will break down each step, ensuring clarity and comprehension for anyone looking to enhance their algebra skills. This guide aims to provide a comprehensive understanding, making it easier for students and enthusiasts alike to tackle similar problems with confidence. By the end of this article, you will have a firm grasp on how to substitute values into algebraic expressions and simplify them correctly. Let’s embark on this algebraic journey together and unravel the solution to this problem.
Understanding the Basics of Algebraic Expressions
Before we dive into the solution, it's important to understand the basics of algebraic expressions. An algebraic expression is a combination of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. In our expression, 2x + 3(x - 11), x is the variable, 2 and 3 are coefficients, and 11 is a constant. The operations involved are multiplication (between the coefficient and the variable) and addition/subtraction. To solve an algebraic expression for a specific value of the variable, we substitute the given value for the variable and then simplify the expression using the order of operations, often remembered by the acronym PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). This foundational knowledge is essential for accurately evaluating any algebraic expression. Understanding the role of each component – variables, coefficients, constants, and operations – sets the stage for a smooth problem-solving process. Moreover, recognizing the importance of the order of operations ensures that we arrive at the correct answer, maintaining the integrity of the mathematical equation. With a solid grasp of these fundamentals, we can confidently approach the task of solving our expression.
Step-by-Step Solution
Step 1: Substitute the Value of x
The first step in solving the expression 2x + 3(x - 11) when x = 6 is to substitute the value of x into the expression. This means replacing every instance of x with the number 6. So, the expression becomes:
2(6) + 3(6 - 11)
This substitution is a crucial initial step because it transforms the algebraic expression into an arithmetic one, which we can then simplify using the order of operations. It's essential to perform this step carefully to avoid any errors that could propagate through the rest of the solution. Accurate substitution ensures that we are working with the correct numerical values, setting the stage for a reliable final answer. By replacing the variable x with its given value, we bridge the gap between abstract algebra and concrete arithmetic, making the problem more approachable and manageable. This foundational step lays the groundwork for the subsequent calculations and is a key element in the successful resolution of the algebraic expression.
Step 2: Simplify Inside the Parentheses
Following the order of operations (PEMDAS/BODMAS), we must first address the operations inside the parentheses. In our expression, we have (6 - 11) inside the parentheses. Performing this subtraction, we get:
6 - 11 = -5
So, our expression now looks like this:
2(6) + 3(-5)
Simplifying the parentheses is a critical step because it reduces the complexity of the expression and prepares it for further operations. It ensures that we are dealing with simplified terms, making the subsequent multiplication and addition steps more straightforward. Accurate subtraction within the parentheses is essential for maintaining the integrity of the equation and ensuring that the final result is correct. By prioritizing this step, we adhere to the fundamental principles of order of operations, which are crucial for solving algebraic expressions effectively. This careful simplification lays a solid foundation for the remaining calculations and contributes significantly to the overall accuracy of the solution.
Step 3: Perform the Multiplications
Next, we need to perform the multiplications. We have two multiplications in our expression: 2(6) and 3(-5). Let's calculate these:
- 2(6) = 12
- 3(-5) = -15
Now, our expression looks like this:
12 + (-15)
Performing the multiplications is a vital step in simplifying the expression because it reduces the number of terms and prepares the expression for the final addition. Accurate multiplication is crucial to avoid errors that could affect the final result. By addressing the multiplications in the correct order, we adhere to the rules of algebraic manipulation and ensure that the solution progresses logically and accurately. This step transforms the expression into a simpler form, making it easier to manage and leading us closer to the final answer. Careful and precise multiplication is key to maintaining the integrity of the equation and ensuring the overall correctness of the solution.
Step 4: Perform the Addition
Finally, we perform the addition: 12 + (-15). Adding a negative number is the same as subtracting its positive counterpart, so we have:
12 - 15 = -3
Therefore, the value of the expression 2x + 3(x - 11) when x = 6 is -3.
Performing the addition is the final step in simplifying the expression and arriving at the solution. Accurate addition (or subtraction in this case) is crucial for obtaining the correct answer. This step consolidates all the previous calculations into a single, final value, representing the result of the entire algebraic expression for the given value of x. By carefully executing this final operation, we complete the process of solving the algebraic problem and provide a definitive answer. The precision and attention to detail in this step ensure that the solution is both accurate and reliable, marking the successful completion of the problem-solving process.
Conclusion
In conclusion, we have successfully solved the algebraic expression 2x + 3(x - 11) when x = 6. By following the order of operations and breaking down the problem into smaller, manageable steps, we arrived at the solution -3. This process involved substituting the value of x, simplifying inside the parentheses, performing the multiplications, and finally, performing the addition. This example demonstrates the importance of understanding the fundamentals of algebra and applying them systematically. Mastering these concepts is essential for tackling more complex algebraic problems in the future. The step-by-step approach not only helps in solving the problem accurately but also enhances comprehension of the underlying algebraic principles. By practicing similar problems, one can build confidence and proficiency in algebra, which is a valuable skill in various fields of study and real-life applications. The ability to solve algebraic expressions efficiently and accurately is a testament to one's mathematical acumen and problem-solving capabilities.
This comprehensive guide has provided a clear and concise method for solving algebraic expressions. Remember, practice is key to mastering algebra. Keep practicing, and you'll become more confident and proficient in solving these types of problems.