Evaluate The Expression 4a + 5(a - B) + 6b + 3 When A = 2 And B = 6

In the realm of mathematics, evaluating expressions is a fundamental skill. It's like deciphering a code, where you're given a set of instructions (the expression) and specific values for the variables involved. Your mission? To follow the instructions, substitute the values, and arrive at a single numerical answer. In this article, we'll break down the process of evaluating expressions, focusing on the expression 4a + 5(a - b) + 6b + 3 when a = 2 and b = 6. We'll take a step-by-step approach, ensuring you understand each stage of the process.

Understanding the Basics

Before we dive into the expression, let's clarify some key terms. An expression is a combination of numbers, variables, and mathematical operations (+, -, ×, ÷). Variables are symbols (usually letters like a, b, x, or y) that represent unknown values. To evaluate an expression, we replace the variables with their given values and perform the operations according to the order of operations (PEMDAS/BODMAS). PEMDAS is a mnemonic device that helps us remember the order of operations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This is a crucial concept because the order in which you perform operations dramatically affects the result. Without a consistent set of rules, mathematical expressions could be interpreted in multiple ways, leading to chaos and confusion. For instance, consider the simple expression 2 + 3 * 4. If we perform the addition first, we get 5 * 4 = 20. But if we follow PEMDAS and perform the multiplication first, we get 2 + 12 = 14. The second answer is the correct one, highlighting the importance of adhering to the established order of operations. In more complex expressions involving multiple sets of parentheses or nested operations, breaking down the problem into smaller, manageable steps is often the best approach. This involves carefully identifying the innermost operations and working your way outwards, ensuring that you correctly apply PEMDAS at each stage. With practice and a systematic approach, evaluating expressions becomes a seamless and intuitive process. It's not just about crunching numbers; it's about understanding the structure of mathematical language and expressing relationships in a clear and unambiguous way.

Step 1: Substitution

The first step in evaluating the expression is to substitute the given values for the variables. In our case, we have a = 2 and b = 6. So, we replace every instance of a with 2 and every instance of b with 6 in the expression. This gives us: 4(2) + 5(2 - 6) + 6(6) + 3. Substitution is the bedrock upon which the entire evaluation process rests. It's the act of replacing abstract symbols (the variables) with concrete numerical values, transforming a symbolic expression into a numerical one that we can actually calculate. Accuracy in this initial stage is paramount; a single incorrect substitution can throw off the entire calculation, leading to a wrong answer. Think of it like building a house – a faulty foundation will inevitably compromise the entire structure. Similarly, a mistake during substitution will cascade through the subsequent steps, rendering the final result invalid. Therefore, it's essential to double-check each substitution, ensuring that every variable has been correctly replaced with its corresponding value. This meticulous attention to detail will save you from unnecessary errors and frustration down the line. Furthermore, mastering the art of substitution is not just about avoiding mistakes; it's also about developing a deeper understanding of the relationship between variables and their numerical representations. It's about recognizing that variables are placeholders, and that by substituting values, we're essentially giving them a specific meaning within the context of the expression. This understanding is crucial for tackling more complex mathematical problems, where variables might represent physical quantities, financial data, or even abstract concepts. By practicing substitution, we're honing our ability to translate symbolic representations into concrete realities, a skill that is invaluable in various fields of study and real-world applications.

Step 2: Parentheses/Brackets

Following PEMDAS, we tackle the parentheses first. Inside the parentheses, we have (2 - 6), which equals -4. Our expression now becomes: 4(2) + 5(-4) + 6(6) + 3. Parentheses, or brackets, in mathematical expressions act as powerful grouping symbols. They dictate the order in which operations are performed, essentially creating mini-expressions within the larger one. By enclosing a portion of the expression within parentheses, we're telling the mathematical world: "Calculate this part first, before you do anything else." This seemingly simple convention has profound implications for the way we interpret and evaluate expressions. Without parentheses, the order of operations would be strictly governed by the standard PEMDAS/BODMAS rules. However, parentheses give us the flexibility to override these rules and prioritize certain calculations. For instance, consider the expression 3 + 2 * 4. According to PEMDAS, we would perform the multiplication first, resulting in 3 + 8 = 11. But if we introduce parentheses, (3 + 2) * 4, we force the addition to be performed first, yielding 5 * 4 = 20. This demonstrates the dramatic impact that parentheses can have on the final result. When dealing with complex expressions involving multiple sets of parentheses, it's crucial to work from the innermost set outwards. This ensures that you're correctly applying the order of operations at each level of nesting. Imagine a set of Russian nesting dolls – you need to open the outermost doll to reveal the next one, and so on. Similarly, you need to resolve the innermost parentheses before you can move on to the outer ones. Furthermore, parentheses are not just about changing the order of operations; they also play a vital role in clarifying the meaning of an expression. They can help to visually group terms, making it easier to understand the structure and intent of the expression. In essence, parentheses are the punctuation marks of mathematical language, guiding us in how to read, interpret, and evaluate expressions accurately.

Step 3: Multiplication

Next, we perform the multiplications. We have 4(2) = 8, 5(-4) = -20, and 6(6) = 36. Substituting these results into the expression, we get: 8 + (-20) + 36 + 3. Multiplication, in the context of mathematical expressions, is more than just a basic arithmetic operation; it's a powerful tool for scaling, combining, and transforming quantities. It represents repeated addition, but its significance extends far beyond that simple definition. In the order of operations hierarchy, multiplication holds a prominent position, taking precedence over addition and subtraction. This reflects its fundamental role in many mathematical and scientific contexts. Think of multiplication as a way to amplify or diminish a quantity. Multiplying a number by a value greater than 1 increases its magnitude, while multiplying by a value between 0 and 1 reduces it. This scaling property is crucial in various applications, from calculating percentage increases and decreases to determining the effects of levers and gears. Furthermore, multiplication is the cornerstone of many algebraic operations. It's used to combine variables and coefficients, creating terms that represent complex relationships. The distributive property, which governs how multiplication interacts with addition and subtraction, is a fundamental concept in algebra and beyond. It allows us to expand expressions, simplify equations, and solve for unknown variables. In evaluating expressions, performing multiplication before addition or subtraction is essential for obtaining the correct result. This is because multiplication often represents a more tightly coupled relationship between quantities than addition or subtraction. For instance, in the expression 2 + 3 * 4, the 3 * 4 represents a single quantity that should be calculated before it's added to the 2. Failing to adhere to this order would lead to a misinterpretation of the expression and an incorrect answer. Mastering the intricacies of multiplication, therefore, is not just about memorizing multiplication tables; it's about understanding its fundamental role in mathematical reasoning and its power to represent and manipulate quantities in a meaningful way.

Step 4: Addition

Finally, we perform the additions from left to right. 8 + (-20) = -12. Then, -12 + 36 = 24. And lastly, 24 + 3 = 27. So, the value of the expression is 27. Addition, in the realm of mathematics, is the fundamental operation of combining quantities. It's the process of bringing together two or more numbers, variables, or expressions to find their sum. While seemingly simple, addition is the bedrock upon which much of mathematics is built. It's the foundation for more complex operations like subtraction, multiplication, and division, and it plays a crucial role in countless applications across various fields. The order in which we perform addition can sometimes seem arbitrary, but a consistent approach is essential for clarity and accuracy. In the order of operations, addition and subtraction are typically performed from left to right. This convention ensures that expressions are evaluated in a consistent and unambiguous manner. However, the associative property of addition allows us to regroup terms without changing the result. This means that (a + b) + c is equivalent to a + (b + c). This property can be useful for simplifying expressions or performing calculations more efficiently. For instance, if we have the expression 1 + 2 + 3 + 4, we can add the numbers in any order we choose. We could add them sequentially from left to right, or we could pair them up – (1 + 4) + (2 + 3) – to make the calculation easier. Furthermore, addition is not just about combining numbers; it's also about understanding the relationships between them. Adding a positive number increases the value, while adding a negative number decreases it. This concept is crucial for working with signed numbers and for understanding concepts like debt and credit in financial contexts. In evaluating expressions, addition is often the final step, bringing together the results of previous operations to arrive at a single numerical answer. It's the culmination of a process, the point where we synthesize the various components of the expression into a cohesive whole. Mastering addition, therefore, is not just about knowing how to add numbers; it's about understanding its fundamental role in mathematical reasoning and its power to represent and manipulate quantities in a meaningful way.

Final Answer

Therefore, when a = 2 and b = 6, the expression 4a + 5(a - b) + 6b + 3 evaluates to 27.

Evaluating Expressions When a = 2 and b = 6: A Comprehensive Guide

In this comprehensive guide, we've explored the process of evaluating the expression 4a + 5(a - b) + 6b + 3 when a = 2 and b = 6. We've broken down each step, from substitution to the final calculation, emphasizing the importance of following the order of operations (PEMDAS/BODMAS). Evaluating expressions is a crucial skill in mathematics, and by understanding the underlying principles, you can confidently tackle more complex problems. Remember, practice makes perfect, so keep working on different expressions to hone your skills. If you have any further questions, don't hesitate to seek help from your teacher or a tutor. Mathematics is a journey of discovery, and with each problem you solve, you're building a stronger foundation for future success. The ability to evaluate expressions is not just a mathematical skill; it's a critical thinking skill that can be applied in various fields. It's about breaking down complex problems into smaller, manageable steps, and following a logical process to arrive at a solution. This is a valuable skill in any profession, from engineering to finance to computer science. By mastering the art of evaluating expressions, you're not just learning math; you're developing the ability to think critically and solve problems effectively.