Evaluate The Expression Ab + Xy + Bb When A = 3, B = 2, X = 4, And Y = X. What Is The Final Value?
Introduction
In this article, we will delve into the process of evaluating a given algebraic expression by substituting specific numerical values for the variables involved. Algebraic expressions form the foundation of mathematical problem-solving, and the ability to accurately evaluate them is a crucial skill. We will focus on the expression ab + xy + bb, where the variables are assigned the following values: a = 3, b = 2, x = 4, and y = x. By systematically substituting these values and performing the arithmetic operations, we will arrive at the final numerical value of the expression. This exercise not only reinforces the order of operations but also highlights the importance of precision in mathematical calculations. Understanding the step-by-step evaluation process is essential for tackling more complex algebraic problems and applications in various fields such as physics, engineering, and computer science.
Understanding the Expression
The expression we aim to evaluate is ab + xy + bb. This is a combination of algebraic terms, each involving multiplication and addition. Let's break down the components to ensure clarity before we proceed with the substitution and calculation. The first term, ab, represents the product of the variables a and b. In algebra, when two variables are written side by side without an explicit operator, it implies multiplication. Similarly, the term xy represents the product of the variables x and y. The last term, bb, might look a bit different, but it's simply another way of writing b squared, or b multiplied by itself. Understanding this notation is crucial for accurately substituting the values and performing the calculations. The addition signs between the terms indicate that we need to sum up the results of each multiplication. This expression is a fundamental example of how algebraic terms can be combined to form more complex mathematical statements. Mastering the evaluation of such expressions is a stepping stone to understanding and manipulating more advanced algebraic concepts.
Step-by-Step Evaluation
To evaluate the expression ab + xy + bb with the given values a = 3, b = 2, x = 4, and y = x, we will follow a systematic, step-by-step approach. This method ensures accuracy and clarity in our calculations. First, we substitute the given values into the expression. Since y = x, we know that y = 4. Now our expression looks like this: (3)(2) + (4)(4) + (2)(2). The parentheses are used here to emphasize the multiplication taking place between the variables. Next, we perform the multiplication operations according to the order of operations (PEMDAS/BODMAS), which prioritizes multiplication before addition. The first term, (3)(2), equals 6. The second term, (4)(4), equals 16. The third term, (2)(2), equals 4. Now our expression is simplified to: 6 + 16 + 4. Finally, we perform the addition operations from left to right. 6 plus 16 equals 22, and 22 plus 4 equals 26. Therefore, the final value of the expression ab + xy + bb when a = 3, b = 2, x = 4, and y = x is 26. This step-by-step evaluation demonstrates the importance of following the correct order of operations to arrive at the accurate result.
Step 1: Substitute the Values
The initial and perhaps most critical step in evaluating any algebraic expression is the accurate substitution of the given values for the variables. In our case, the expression is ab + xy + bb, and we are given the values a = 3, b = 2, x = 4, and y = x. This means that wherever we see the variable a in the expression, we replace it with the number 3. Similarly, we replace b with 2 and x with 4. An important point to note here is that y = x, which means the value of y is the same as the value of x. Therefore, we also replace y with 4. After the substitution, the expression transforms from a symbolic representation to a numerical one. It now reads: (3)(2) + (4)(4) + (2)(2). This transformation is a fundamental step in bridging the gap between abstract algebra and concrete numerical calculation. The use of parentheses here is deliberate; it clearly indicates that the numbers are being multiplied. Ensuring correct substitution is paramount, as any error at this stage will propagate through the rest of the calculation, leading to an incorrect final answer. This step emphasizes the importance of attention to detail and careful transcription in mathematical problem-solving.
Step 2: Perform Multiplication
Once we have successfully substituted the numerical values for the variables, the next step is to perform the multiplication operations. In the expression (3)(2) + (4)(4) + (2)(2), we have three multiplication operations to carry out. According to the order of operations (PEMDAS/BODMAS), multiplication takes precedence over addition. This means we must compute the products before we can sum the terms. Let's start with the first term, (3)(2). This simply means 3 multiplied by 2, which equals 6. Next, we move on to the second term, (4)(4). This is 4 multiplied by itself, which equals 16. Finally, we calculate the third term, (2)(2). This is 2 multiplied by itself, which equals 4. After performing these multiplications, the expression now looks like this: 6 + 16 + 4. We have effectively simplified the expression by reducing the multiplication terms to single numerical values. This step highlights the importance of adhering to the order of operations, as performing addition before multiplication would lead to a completely different result. Accurate multiplication is crucial for arriving at the correct final answer, and this step serves as a building block for the subsequent addition operations.
Step 3: Perform Addition
After completing the multiplication operations, we arrive at the final step: performing the addition. Our expression has been simplified to 6 + 16 + 4. Now, we need to add these numbers together to find the final value of the expression. Addition is a straightforward operation, but it's important to perform it systematically to avoid errors. We typically perform addition from left to right. So, we start by adding the first two numbers: 6 plus 16. This gives us a sum of 22. Now, we have the expression 22 + 4. We add the remaining number, 4, to 22. This gives us a final sum of 26. Therefore, the value of the expression ab + xy + bb when a = 3, b = 2, x = 4, and y = x is 26. This final addition step consolidates all the previous calculations and provides us with the numerical answer we were seeking. It underscores the cumulative nature of mathematical operations, where each step builds upon the previous ones to arrive at the solution. This result, 26, is the final evaluated value of the given algebraic expression for the specified variable values.
Final Answer
After meticulously following the step-by-step evaluation process, we have arrived at the final answer. The expression ab + xy + bb, when evaluated with the values a = 3, b = 2, x = 4, and y = x (which means y = 4), equals 26. This result is the culmination of substituting the given values, performing the necessary multiplication operations, and then adding the resulting terms. Each step was crucial in ensuring the accuracy of the final answer. The substitution transformed the abstract algebraic expression into a concrete numerical calculation. The multiplication operations simplified the expression further, and the final addition consolidated the results into a single numerical value. This entire process demonstrates the power and precision of algebraic evaluation. Understanding how to correctly evaluate expressions is fundamental to algebra and has wide-ranging applications in various fields of mathematics and beyond. The final answer, 26, represents the quantitative value of the expression under the given conditions, and it serves as a testament to the methodical approach we employed.
Conclusion
In conclusion, the evaluation of the algebraic expression ab + xy + bb with the specified values a = 3, b = 2, x = 4, and y = x has provided a valuable exercise in algebraic manipulation and order of operations. We began by understanding the expression itself, breaking it down into its constituent terms and recognizing the implied multiplication between variables. The crucial first step involved substituting the given numerical values for the variables, transforming the expression into a purely numerical one. We then meticulously performed the multiplication operations, adhering to the correct order of operations (PEMDAS/BODMAS). This resulted in a simplified expression ready for the final addition step. Adding the resulting terms together, we arrived at the final answer: 26. This process highlights the importance of a systematic and methodical approach to mathematical problem-solving. Each step, from substitution to multiplication and finally addition, played a critical role in achieving the accurate result. The ability to evaluate algebraic expressions is a fundamental skill in mathematics, with applications spanning various disciplines. This exercise serves as a solid foundation for tackling more complex algebraic problems and underscores the significance of precision and attention to detail in mathematical calculations.