Mathematical Law Mastery: Solving Equations Step-by-Step

Hey math enthusiasts! Today, we're diving into a cool problem that explores a specific mathematical law. We'll break it down step-by-step, making sure everyone understands the process. So, let's get started! This exercise revolves around a newly defined operation and it's a fantastic way to sharpen those problem-solving skills. The key to conquering this type of problem lies in understanding the given law and applying it meticulously. We'll be working with real numbers and a special operation denoted by the asterisk (*). This operation isn't your standard multiplication, but a custom rule. The law states: x * y = (2x - 1)(2y - 1) + 1. This means that whenever we see this operation between two numbers, we have to apply this specific formula. Remember, understanding this initial law is crucial, so let's make sure we're all on the same page. The first part of the problem asks us to demonstrate that 1 * 2 = 4. This seems straightforward, but it's a great way to get comfortable with the new operation. Let's jump right in and see how it all works. We'll carefully substitute the given values (x = 1 and y = 2) into our formula and calculate the result. This simple step will help us gain confidence in the process, before moving on to more complicated examples. Stick with me, and by the end, you'll be an expert at applying this law! You'll notice that it's just like following a recipe, but with numbers.

Demonstrating 1 * 2 = 4

Alright, let's show that 1 * 2 = 4 using the mathematical law we were given. This is the first step, and it helps us get familiar with the formula. Remember that the rule says x * y = (2x - 1)(2y - 1) + 1. Now, let's substitute x with 1 and y with 2. So, we get 1 * 2 = (2 * 1 - 1)(2 * 2 - 1) + 1. If we simplify the equation carefully, we first handle the terms inside the parentheses: (2 * 1 - 1) becomes (2 - 1), which equals 1. And, (2 * 2 - 1) becomes (4 - 1), which equals 3. So, the equation now reads as 1 * 2 = (1)(3) + 1. Next, we multiply 1 by 3, which gives us 3. Then we add 1, so we get 3 + 1 = 4. Consequently, we have demonstrated that 1 * 2 indeed equals 4. This first part is designed to build your confidence and make sure that everyone can do basic calculations with the new mathematical law. This kind of practice is essential before moving on to more complex aspects of the problem. Always double-check your steps and make sure you are following the order of operations, as it makes a huge difference.

Finding the Real Numbers x Where x * x = 2

Now, let's get into the next part! We need to find the real numbers x that satisfy the equation x * x = 2. This is a bit more interesting, since we are now dealing with an unknown variable. The core idea remains the same: use the law we are given. Remember the law: x * y = (2x - 1)(2y - 1) + 1. Now, let's apply this law with x * x. This means we are replacing y with x. So the formula becomes x * x = (2x - 1)(2x - 1) + 1. To simplify this further, we can write (2x - 1)(2x - 1) as (2x - 1)^2. Thus, our equation becomes x * x = (2x - 1)^2 + 1. Now, we want to find the values of x that make x * x equal to 2. So, we set the equation equal to 2: (2x - 1)^2 + 1 = 2. Let's start by isolating the squared term. Subtracting 1 from both sides, we get (2x - 1)^2 = 1. This is where things get exciting! Now we need to take the square root of both sides. The square root of 1 can be either 1 or -1. So we have two possible scenarios to consider: first, 2x - 1 = 1. Adding 1 to both sides, we get 2x = 2. Dividing by 2, we find that x = 1. Second, 2x - 1 = -1. Adding 1 to both sides, we get 2x = 0. Dividing by 2, we find that x = 0. Consequently, the two real numbers x that satisfy the equation x * x = 2 are x = 1 and x = 0. It's always a good idea to check these solutions by plugging them back into the original equation. This step makes sure there are no errors.

Determining the Non-Zero Integer m Where m * (1 + 1/m) = 1

Alright, let's solve the final part! This one asks us to determine the non-zero integer m that satisfies the equation m * (1 + 1/m) = 1. This part combines what we have already learned with some algebra. Let's use the formula x * y = (2x - 1)(2y - 1) + 1, and replace x with m and y with (1 + 1/m). So the equation will be: m * (1 + 1/m) = (2m - 1)(2(1 + 1/m) - 1) + 1. We know that m * (1 + 1/m) = 1, so we can write: 1 = (2m - 1)(2(1 + 1/m) - 1) + 1. Now, let's simplify the terms. Subtract 1 from both sides, and we get: 0 = (2m - 1)(2 + 2/m - 1). Which further simplifies to: 0 = (2m - 1)(1 + 2/m). We now need to solve for m. Since the product is zero, at least one of the factors must be zero. Let's check these. Case 1: 2m - 1 = 0. Solving for m, we get 2m = 1, which means m = 1/2. But we were asked to find an integer, so this is not a valid solution. Case 2: 1 + 2/m = 0. Subtracting 1 from both sides, we get 2/m = -1. Multiplying both sides by m, we get 2 = -m. Therefore, m = -2. Since -2 is an integer, and it's not zero, this is the correct solution. So, the non-zero integer m that satisfies the equation m * (1 + 1/m) = 1 is m = -2. To be absolutely sure, always verify your result by substituting it back into the original equation. This not only helps prevent mistakes but also improves your understanding of the problem. This concludes our journey through this mathematical law problem. Keep practicing and have fun!