Solve For M: Complex Number Equation

Hey math enthusiasts! Let's dive into a fun little problem today involving complex numbers. We're going to figure out the value of 'm' in this equation: $(9 - 6i) \times m = 9 - 6i$. Don't worry, it's not as scary as it looks. We'll break it down step by step, so even if you're new to complex numbers, you'll be able to follow along. This is a classic example of how we can manipulate equations to isolate a variable, and it's a fundamental concept in algebra. Let's get started, shall we?

Understanding the Problem: The Equation and Complex Numbers

Alright, guys, let's start with the basics. We're given the equation $(9 - 6i) \times m = 9 - 6i$. What does this even mean? Well, we have a complex number, $(9 - 6i)$, being multiplied by 'm', and the result is the same complex number. Complex numbers, for those who might need a refresher, are numbers that can be written in the form $a + bi$, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as the square root of -1. So, in our equation, we have $(9 - 6i)$, where 9 is the real part and -6 is the imaginary part. We need to find the value of 'm' that makes this equation true. This problem is essentially asking: "What number do we multiply $(9 - 6i)$ by to get $(9 - 6i)$?" It's kind of like asking what number you multiply by 5 to get 5 – the answer is, of course, 1. The key is understanding that 'i' behaves in certain ways, but in this particular problem, we don't even need to worry too much about the 'i'. The structure of the equation gives us a pretty big hint about the solution. Think of it like this: If something multiplied by 'm' equals itself, what is 'm' equal to? The properties of multiplication are key here, particularly the identity property of multiplication. This property states that any number multiplied by 1 equals itself. This will be very helpful for solving this equation.

Solving for 'm': Step-by-Step Explanation

Okay, let's get down to the nitty-gritty and solve for 'm'. As we mentioned before, the equation is $(9 - 6i) \times m = 9 - 6i$. To isolate 'm', we need to think about what operation is being done to it. In this case, $(9 - 6i)$ is multiplying 'm'. To undo multiplication, we use division. So, we'll divide both sides of the equation by $(9 - 6i)$. This gives us:

$m = \frac{9 - 6i}{9 - 6i}$.

See how the complex number on both sides is cancelled? Now, any non-zero number divided by itself equals 1. Therefore, $m = 1$. It’s as simple as that! We didn't even have to do any complex calculations with the imaginary unit 'i'. The equation was set up in a way that the complex part cancels itself, leaving us with a straightforward solution. So, the value of 'm' is 1. We're not getting any of the options A, C, or D; hence, the correct answer is the one that resembles '1'. This method of solving equations is fundamental across all branches of mathematics. By understanding this approach, you can easily solve similar problems. Moreover, remember that it's important to always check your answers to ensure the solution is correct.

Checking the Answer and Understanding the Choices

Let's check our answer to make sure we're on the right track, yeah? We found that $m = 1$. Let's plug that back into the original equation: $(9 - 6i) \times 1 = 9 - 6i$. Yup, that checks out! Our answer is correct. Now, let's look at the multiple-choice options and see why the other answers are wrong:

A. $-9 + 6i$: If $m = -9 + 6i$, then $(9 - 6i) \times (-9 + 6i)$ would not equal $(9 - 6i)$. Multiplying this out would result in a different complex number altogether. Therefore, this option is incorrect. It appears that this choice probably stems from confusing the concept of additive inverses. Remember, we are not looking for a number that when added to $9-6i$ gives us 0. We're looking for a number that multiplies it and results in $9 - 6i$.

B. i: If $m = i$, then $(9 - 6i) \times i$ would not equal $(9 - 6i)$. Multiplying this out would give a different complex number. The imaginary unit 'i' has a specific set of rules when multiplied by other complex numbers. So, this option is also incorrect.

C. $9 + 6i$: If $m = 9 + 6i$, then $(9 - 6i) \times (9 + 6i)$ would not equal $(9 - 6i)$. Multiplying this out would give a real number (81 + 36 = 117), not the original complex number. This option is not correct because it changes the imaginary part's sign without addressing the equation's core requirement. When solving equations, understanding how to apply the correct mathematical operations is crucial.

D. 0: If $m = 0$, then $(9 - 6i) \times 0 = 0$, which is not equal to $(9 - 6i)$. This option would result in a zero product, which does not satisfy the original equation. Therefore, this answer is also wrong.

Therefore, by process of elimination and by actually working out the equation, the correct choice is not given, as '1' is the correct answer and is not available in the given options.

Conclusion: Mastering the Equation

So there you have it, guys! We've successfully solved for 'm' in the equation $(9 - 6i) \times m = 9 - 6i$. The answer is $m = 1$. This problem was a straightforward application of the identity property of multiplication. It demonstrates how fundamental mathematical concepts can be applied to complex numbers. Remember, the key is to understand the properties of the numbers and operations involved. Keep practicing, keep learning, and you'll become a math whiz in no time! Always remember to double-check your work, and don't be afraid to break down problems into smaller, more manageable steps. By understanding how to solve simple equations like this, you're building a strong foundation for tackling more complex mathematical problems in the future. Keep up the awesome work, and keep exploring the fascinating world of mathematics!