Solving The Operation: A * B = (a-b)/(a + 2b)
Introduction
In this article, we'll dive into a specific mathematical operation defined as a * b = (a - b) / (a + 2b). This operation isn't your standard addition, subtraction, multiplication, or division; it's a unique rule that combines these basic operations in a particular way. Our main goal is to understand how this operation works and then apply it to find the values of two specific calculations: 1 * 2 and -4 * 2. By working through these examples, you'll gain a clearer understanding of how to handle new or unfamiliar mathematical operations. We'll break down each step, making it easy to follow along, and ensure you grasp the core concept behind this operation. So, whether you're a student brushing up on your math skills or simply curious about different mathematical expressions, this guide will help you navigate through the process with confidence.
Understanding the Operation
The heart of this problem lies in understanding the defined operation: a * b = (a - b) / (a + 2b). This equation tells us exactly how to combine two numbers, a and b, using this particular operation. It's essential to recognize that this isn't a standard mathematical operation like addition or multiplication, which have well-established properties and rules. Instead, it's a custom-defined operation, which means we must adhere strictly to the given formula. The operation involves subtracting b from a in the numerator and adding a to twice b in the denominator. The result is the quotient of these two expressions. To effectively use this operation, we need to carefully substitute the given values for a and b into the equation. It’s also crucial to pay close attention to the order of operations (PEMDAS/BODMAS), ensuring that we perform the subtraction and addition correctly before carrying out the division. This understanding forms the foundation for solving the specific calculations that follow. Remember, every operation, no matter how unique, has a precise definition, and adhering to that definition is key to finding the correct result.
Calculating 1 * 2
Let's begin by calculating the value of 1 * 2 using the defined operation a * b = (a - b) / (a + 2b). In this case, a equals 1 and b equals 2. The first step is to substitute these values into the equation. This means replacing every instance of a with 1 and every instance of b with 2. When we perform this substitution, the equation becomes (1 - 2) / (1 + 2 * 2). Now, we need to simplify the expression following the order of operations (PEMDAS/BODMAS). First, we tackle the operations inside the parentheses. In the numerator, 1 - 2 equals -1. In the denominator, we have 1 + 2 * 2. According to the order of operations, we perform multiplication before addition, so 2 * 2 equals 4. Then, we add 1 to 4, which gives us 5. Thus, the expression simplifies to -1 / 5. This fraction represents the final value of 1 * 2 under this operation. The calculation demonstrates the application of the defined operation, emphasizing the importance of accurate substitution and adherence to the order of operations.
Calculating -4 * 2
Next, we will calculate the value of -4 * 2 using the same operation, a * b = (a - b) / (a + 2b). This time, a equals -4 and b equals 2. Similar to the previous calculation, the first step is to substitute these values into the equation. Replacing a with -4 and b with 2, the equation becomes (-4 - 2) / (-4 + 2 * 2). Now, we need to simplify this expression following the order of operations. Starting with the numerator, -4 - 2 equals -6. In the denominator, we have -4 + 2 * 2. Again, we perform multiplication before addition, so 2 * 2 equals 4. The expression in the denominator then becomes -4 + 4, which equals 0. So, the expression simplifies to -6 / 0. Here, we encounter a critical issue: division by zero. In mathematics, division by zero is undefined because it leads to an indeterminate result. There's no number that, when multiplied by zero, gives a non-zero number. Therefore, the value of -4 * 2 under this operation is undefined. This calculation highlights an important consideration when working with mathematical operations: the potential for undefined results, especially when dealing with fractions and division.
Conclusion
In conclusion, we have explored a unique mathematical operation defined as a * b = (a - b) / (a + 2b). We successfully calculated the value of 1 * 2, which resulted in -1/5. This calculation demonstrated the direct application of the operation by substituting the given values and following the order of operations. However, when we attempted to calculate -4 * 2, we encountered a situation where the result was undefined due to division by zero. This underscores a crucial point in mathematics: not all operations are defined for all possible inputs. Division by zero is a fundamental limitation that must always be considered. Through these examples, we've learned how to apply a custom-defined operation and the importance of being mindful of potential undefined results. Understanding these concepts allows for a more comprehensive approach to mathematical problem-solving, especially when dealing with new or unfamiliar operations. For further exploration of mathematical operations and problem-solving techniques, you might find valuable resources on websites like Khan Academy, which offers a wide range of math tutorials and exercises.
