Evaluating The Expression (3x - 12) + (1/2)(xy - 10) For X = 4 And Y = 6

Introduction

In this article, we will delve into the process of evaluating algebraic expressions by substituting given values for variables. Specifically, we will focus on finding the value of the expression $(3x - 12) + \frac{1}{2}(xy - 10)$ when $x = 4$ and $y = 6$. This type of problem is a fundamental concept in algebra, often encountered in middle and high school mathematics. Understanding how to correctly substitute values and simplify expressions is crucial for mastering more advanced algebraic topics. We will break down the process step-by-step, ensuring clarity and comprehension for readers of all mathematical backgrounds. The key to successfully solving these types of problems lies in following the order of operations (PEMDAS/BODMAS) and careful substitution. This article aims to not only provide the solution but also to illustrate the underlying principles and techniques involved in evaluating algebraic expressions.

Understanding Algebraic Expressions

Before we dive into the specific problem, let's first define what an algebraic expression is. An algebraic expression is a combination of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Variables are symbols (usually letters) that represent unknown values, while constants are fixed numerical values. For example, in the expression $(3x - 12) + \frac{1}{2}(xy - 10)$, $x$ and $y$ are variables, and 3, 12, and $\frac{1}{2}$ are constants. The operations involved are multiplication, subtraction, and addition. Understanding the structure of an algebraic expression is essential for evaluating it correctly. Each term in the expression must be treated according to the order of operations. Parentheses or brackets often indicate the order in which operations should be performed, with expressions inside parentheses evaluated first. This is a crucial aspect of simplifying and evaluating algebraic expressions, as the order in which operations are performed can significantly impact the final result. Furthermore, recognizing like terms and combining them can simplify the expression before substituting values, although in this particular problem, substitution is the most straightforward approach. Understanding these fundamentals sets the stage for efficiently and accurately evaluating expressions.

Step-by-Step Solution

To find the value of the expression $(3x - 12) + \frac{1}{2}(xy - 10)$ for $x = 4$ and $y = 6$, we will follow these steps:

1. Substitute the given values of $x$ and $y$ into the expression: Replace $x$ with 4 and $y$ with 6 in the expression: $(3(4) - 12) + \frac{1}{2}((4)(6) - 10)$

2. Simplify the expression inside the parentheses: First, evaluate the expressions within the parentheses: $(12 - 12) + \frac{1}{2}(24 - 10)$

3. Continue simplifying within the parentheses: Perform the subtractions inside the parentheses: $(0) + \frac{1}{2}(14)$

4. Perform the multiplication: Multiply $\frac{1}{2}$ by 14: $0 + 7$

5. Finally, perform the addition: Add 0 and 7: $7$

Therefore, the value of the expression $(3x - 12) + \frac{1}{2}(xy - 10)$ for $x = 4$ and $y = 6$ is 7.

Detailed Breakdown of Each Step

Let's break down each step in more detail to ensure a clear understanding of the solution process. The first step, substitution, is crucial. It involves replacing the variables in the expression with their given numerical values. In our case, we replace $x$ with 4 and $y$ with 6. This transforms the algebraic expression into a numerical expression, which can then be simplified. It's important to substitute the values correctly, paying attention to the placement of the numbers and the operations involved. A common mistake is to misplace a number or forget a multiplication sign. Next, we move on to simplification. This involves performing the mathematical operations in the correct order. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), dictates the sequence in which operations should be performed. In our expression, we first simplify the expressions inside the parentheses. This involves performing multiplication and subtraction within the parentheses. After simplifying the parentheses, we perform any remaining multiplications or divisions, followed by additions and subtractions. By meticulously following this order, we ensure that we arrive at the correct value of the expression. Each step in the simplification process builds upon the previous one, leading us to the final solution.

Common Mistakes to Avoid

When evaluating algebraic expressions, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate solutions. One of the most frequent errors is not following the order of operations correctly. For example, some might perform addition before multiplication, leading to an incorrect result. Remember PEMDAS/BODMAS and adhere to it strictly. Another common mistake is incorrect substitution. This can involve substituting the wrong value for a variable or misplacing a number during the substitution process. Double-check your substitutions to ensure they are accurate. Sign errors are also a frequent source of mistakes. Pay close attention to negative signs and ensure they are applied correctly throughout the simplification process. For instance, when distributing a negative sign across parentheses, remember to change the signs of all terms inside the parentheses. Arithmetic errors, such as mistakes in multiplication or addition, can also lead to incorrect answers. To minimize these errors, take your time, show your work clearly, and double-check your calculations. Finally, forgetting to distribute a factor across all terms within parentheses is a common error. If there is a number or variable multiplying a set of parentheses, make sure to multiply it by each term inside the parentheses. By being mindful of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy in evaluating algebraic expressions.

Practice Problems

To further solidify your understanding of evaluating algebraic expressions, it's helpful to work through additional practice problems. Here are a few examples you can try:

1. Evaluate $2x^2 + 3y - 5$ for $x = 2$ and $y = -1$.
2. Find the value of $\frac{4a - b}{2} + c$ when $a = 3$, $b = -2$, and $c = 5$.
3. Calculate the value of $(m + n)^2 - 3mn$ for $m = -4$ and $n = 1$.
4. Determine the value of $5(p - q) + \frac{1}{3}(p + q)$ when $p = 6$ and $q = 3$.

Working through these problems will give you valuable experience in applying the concepts and techniques discussed in this article. Remember to follow the order of operations carefully and double-check your work to avoid errors. The more you practice, the more confident and proficient you will become in evaluating algebraic expressions. Practice helps reinforce the steps involved and allows you to identify and correct any misunderstandings. It's also beneficial to compare your solutions with the correct answers to identify any mistakes and learn from them. By engaging in regular practice, you can develop a strong foundation in algebra and improve your problem-solving skills.

Conclusion

In conclusion, we have successfully evaluated the algebraic expression $(3x - 12) + \frac{1}{2}(xy - 10)$ for $x = 4$ and $y = 6$. By carefully substituting the given values and following the order of operations, we found the value of the expression to be 7. This process illustrates the fundamental principles of evaluating algebraic expressions, which are essential for success in algebra and beyond. We also discussed common mistakes to avoid and provided practice problems to further enhance your understanding. Mastering these skills will enable you to confidently tackle more complex algebraic problems and build a strong foundation in mathematics. Remember, practice is key to proficiency, so continue to work through examples and apply the techniques you've learned. With consistent effort, you can develop a solid understanding of algebraic expressions and their evaluation.