Real Number Condition: Find X And Y For (4 + 5i)(x + Yi)

by Andrew McMorgan 57 views

Hey guys! Ever wondered how complex numbers can sometimes result in good ol' real numbers? It's a cool concept in mathematics, and today, we're diving into a problem that explores just that. We're going to figure out which values of x and y will make the expression (4 + 5i)(x + yi) represent a real number. Let's break it down together!

Understanding Complex Numbers and Real Numbers

Before we jump into solving the problem, let's quickly recap what complex and real numbers are all about. This foundational knowledge is super important for grasping the core of the question and arriving at the correct answer. So, grab your thinking caps, and let’s make sure we’re all on the same page!

First off, a real number is any number that can be plotted on a number line. Think of integers (like -2, 0, 5), rational numbers (like 1/2, 3.75), and irrational numbers (like √2, π). They're the numbers we usually deal with in everyday math. On the other hand, a complex number is a number that can be expressed 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. The a part is the real part, and the bi part is the imaginary part. Complex numbers extend the real number system by including this imaginary unit, allowing us to work with the square roots of negative numbers.

Now, the key to this problem lies in understanding when a complex number becomes a real number. A complex number a + bi is real if its imaginary part is zero, meaning b = 0. In other words, there’s no “i” term hanging around. This is because the imaginary unit i introduces a component that's not on the real number line, so to get a real number, we need to eliminate that imaginary part. This elimination is precisely what we're aiming for in the given expression. We want to find the values of x and y that will make the imaginary part of the result vanish, leaving us with just a real number. This involves some algebraic manipulation and a bit of insight into how complex numbers interact when multiplied.

Expanding the Expression (4 + 5i)(x + yi)

Okay, now comes the fun part – let’s expand the expression (4 + 5i)(x + yi). This is where we put our algebraic skills to the test! We'll use the distributive property (also known as the FOIL method) to multiply these two complex numbers together. Trust me, it’s not as intimidating as it sounds; we'll take it step by step.

Here's how it goes:

  • First, we multiply the first terms in each parenthesis: 4 * x = 4x
  • Then, we multiply the outer terms: 4 * yi = 4yi
  • Next, we multiply the inner terms: 5i * x = 5xi
  • Finally, we multiply the last terms: 5i * yi = 5yi²

So, when we put it all together, we get:

4x + 4yi + 5xi + 5yi²

But wait, we're not done yet! Remember that i is the imaginary unit, and i² = -1. This is a crucial piece of information. Let’s substitute -1 for i² in our expression:

4x + 4yi + 5xi + 5y(-1) = 4x + 4yi + 5xi - 5y

Now, let’s group the real and imaginary parts together. The real parts are the terms without i, and the imaginary parts are the terms with i:

(4x - 5y) + (4y + 5x)i

Ta-da! We've successfully expanded the expression and separated it into its real and imaginary components. Now we have a clear view of what makes up each part, which is essential for the next step in solving our problem. This expanded form is the key to unlocking the values of x and y that will give us a real number.

Setting the Imaginary Part to Zero

Alright, we've reached a critical point in our quest! Remember, for the expression (4 + 5i)(x + yi) to represent a real number, the imaginary part must be equal to zero. This is the golden rule that will guide us to the correct values of x and y. So, let's focus on the imaginary part of our expanded expression, which we found to be (4y + 5x)i. To make this entire term zero, the expression inside the parentheses, (4y + 5x), must be zero.

Therefore, we set up the equation:

4y + 5x = 0

This equation is our ticket to finding the relationship between x and y that satisfies the condition for a real number. We can think of this as a constraint that x and y must adhere to. It tells us that the values of x and y are not independent; they are linked in a way that cancels out the imaginary component when the original expression is evaluated.

Now, we need to solve this equation for one of the variables. It doesn’t matter which one we choose, but let's solve for y to keep things consistent and easy to follow. To isolate y, we first subtract 5x from both sides:

4y = -5x

Then, we divide both sides by 4:

y = (-5/4)x

This equation, y = (-5/4)x, is super important! It tells us that for any value of x we choose, the corresponding value of y must be -5/4 times that value in order for the expression to be a real number. This is the fundamental relationship we've been searching for. Now, with this relationship in hand, we can check the given options and see which pair of x and y values fits the bill.

Checking the Given Options

We've done the hard work of setting up the equation y = (-5/4)x. Now, the final step is to put on our detective hats and check which of the provided options satisfies this equation. This is where we see if our theoretical work aligns with the practical choices given in the problem. Let's go through each option one by one and see if the x and y values fit our relationship.

Option A: x = 4, y = 5

Let's plug these values into our equation: 5 = (-5/4)(4). Simplifying the right side, we get 5 = -5. This is definitely not true! So, Option A is not the correct answer. The values x = 4 and y = 5 do not make the imaginary part vanish.

Option B: x = -4, y = 0

Now, let’s try these values: 0 = (-5/4)(-4). Simplifying, we get 0 = 5. Again, this is not true! Option B is also incorrect. These values do not satisfy our condition for a real number.

Option C: x = 4, y = -5

Plugging these values in: -5 = (-5/4)(4). Simplifying the right side, we get -5 = -5. Bingo! This is true! Option C is a potential solution. The values x = 4 and y = -5 seem to work.

Option D: x = 0, y = 5

Finally, let’s check these values: 5 = (-5/4)(0). Simplifying, we get 5 = 0. This is not true, so Option D is not the answer.

After meticulously checking each option, it's clear that only Option C satisfies our equation y = (-5/4)x. This means that when x = 4 and y = -5, the expression (4 + 5i)(x + yi) will indeed represent a real number. We've successfully navigated the complex world of numbers and found the solution!

The Solution

So, there you have it! The values of x and y that make the expression (4 + 5i)(x + yi) a real number are x = 4 and y = -5. This corresponds to Option C. Wasn't that a fun journey through the land of complex numbers? We started with understanding the basics of real and complex numbers, expanded the given expression, set the imaginary part to zero, and finally, verified our solution by checking the options.

Remember, the key to these types of problems is breaking them down into manageable steps and understanding the underlying principles. In this case, the principle of a complex number becoming real when its imaginary part is zero was crucial. By applying this concept and using our algebraic skills, we were able to solve the problem efficiently and accurately.

Keep practicing and exploring the fascinating world of mathematics, guys! There's always something new and exciting to discover. Until next time, keep those numbers crunching!