Evaluate (2x - 12) + (1/2xy - 10) For X = 8 And Y = 2

Jul 11, 2025 by ADMIN 54 views

Evaluating the Expression (2x - 12) + (1/2xy - 10) for x = 8 and y = 2

Finding the value of algebraic expressions is a fundamental concept in mathematics. In this article, we will walk through the process of evaluating the expression (2x - 12) + (1/2xy - 10) for specific values of x and y. Specifically, we will substitute x = 8 and y = 2 into the expression and simplify it step by step. This process involves applying the order of operations, which is a critical skill in algebra. Understanding how to substitute values into expressions and simplify them is essential for solving more complex mathematical problems. This skill is not only crucial in algebra but also in various fields such as physics, engineering, and computer science, where mathematical models are used extensively. Therefore, mastering this technique is a valuable asset for students and professionals alike. Let's dive into the solution and understand how to evaluate the given expression accurately.

Understanding the Expression

Before we begin the evaluation, let's take a closer look at the given expression:

(2x - 12) + (1/2xy - 10)

This expression consists of two terms enclosed in parentheses, which are then added together. The first term, (2x - 12), involves multiplying the variable x by 2 and then subtracting 12. The second term, (1/2xy - 10), involves multiplying the variables x and y, taking half of the result, and then subtracting 10. The presence of variables indicates that the value of the expression depends on the values of x and y. In this case, we are given specific values for x and y, which allows us to substitute them into the expression and obtain a numerical result. Understanding the structure of the expression is crucial for applying the correct order of operations and simplifying it accurately. Each component of the expression plays a role in determining the final value, and careful attention to each term is necessary to avoid errors. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which we perform the operations to arrive at the correct answer. By following this order, we ensure that the expression is evaluated consistently and accurately.

Step-by-Step Evaluation

Now, let's evaluate the expression (2x - 12) + (1/2xy - 10) for x = 8 and y = 2. Here’s a detailed, step-by-step solution:

1. Substitute the Values

First, we substitute x = 8 and y = 2 into the expression:

(2(8) - 12) + (1/2(8)(2) - 10)

This step involves replacing the variables x and y with their given values. Substitution is a fundamental operation in algebra, allowing us to transform an algebraic expression into a numerical one. Accuracy in substitution is paramount, as any error in this step will propagate through the rest of the solution. It's essential to double-check the values and ensure they are placed correctly in the expression. Once the substitution is complete, we have a purely numerical expression that can be simplified using the order of operations. This process of substitution is not only applicable to simple expressions but also to more complex equations and formulas in various mathematical and scientific contexts. The ability to correctly substitute values is a cornerstone of problem-solving in these fields.

2. Simplify Inside the Parentheses

Next, we simplify the expressions inside the parentheses. According to the order of operations (PEMDAS), we perform multiplication before addition and subtraction.

For the first term, (2(8) - 12):

Multiply 2 by 8: 2 * 8 = 16
Subtract 12 from 16: 16 - 12 = 4

So, the first term simplifies to 4.

For the second term, (1/2(8)(2) - 10):

Multiply 8 by 2: 8 * 2 = 16
Multiply 1/2 by 16: 1/2 * 16 = 8
Subtract 10 from 8: 8 - 10 = -2

So, the second term simplifies to -2.

Simplifying expressions within parentheses is a critical step in evaluating mathematical expressions. This process ensures that we adhere to the correct order of operations and avoid errors. By breaking down each term into smaller, manageable steps, we can systematically simplify the expression. The multiplication operations within the parentheses are performed before the subtraction, following the PEMDAS rule. This meticulous approach to simplification is essential for achieving an accurate final result. The ability to simplify expressions efficiently is a valuable skill in mathematics and is applicable to a wide range of problems. Each step in the simplification process builds upon the previous one, and accuracy at each stage is crucial for the overall correctness of the solution.

3. Add the Simplified Terms

Now, we add the simplified terms together:

4 + (-2) = 4 - 2 = 2

Adding the simplified terms is the final step in evaluating the expression. This step combines the results from the previous simplifications to arrive at the final answer. The addition of a negative number is equivalent to subtraction, so 4 + (-2) becomes 4 - 2. Performing this simple subtraction gives us the final value of the expression. This step highlights the importance of understanding how to work with positive and negative numbers, a fundamental concept in mathematics. Accuracy in this final step is crucial, as it determines the correctness of the entire evaluation. The ability to add numbers, including negative numbers, is a basic yet essential skill that underlies more advanced mathematical operations. The result of this addition provides the numerical value of the expression for the given values of x and y.

Final Result

Therefore, the value of the expression (2x - 12) + (1/2xy - 10) for x = 8 and y = 2 is 2.

Conclusion

In conclusion, we have successfully evaluated the expression (2x - 12) + (1/2xy - 10) for x = 8 and y = 2. By following the order of operations and simplifying each term step by step, we arrived at the final answer of 2. This exercise demonstrates the importance of understanding and applying algebraic principles to solve mathematical problems. The process involved substituting given values for variables, simplifying expressions within parentheses, and performing addition. Each of these steps is a fundamental skill in algebra and is essential for more advanced mathematical concepts. The ability to evaluate expressions accurately is a valuable tool in various fields, including mathematics, science, and engineering. This process not only reinforces basic arithmetic skills but also enhances problem-solving abilities. Mastering these techniques allows individuals to tackle more complex mathematical challenges with confidence. The step-by-step approach used in this solution provides a clear and methodical way to approach similar problems, ensuring accuracy and understanding. By practicing these types of problems, students can develop a strong foundation in algebra and improve their mathematical proficiency.

Answer

The correct answer is 2.