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

by Andrew McMorgan 57 views

Hey Plastik Magazine readers! Ever wondered how complex numbers work and how to make them real? Let's dive into a fascinating problem today that combines complex numbers and a bit of algebraic manipulation. We're going to figure out which values of x and y will turn the expression (4 + 5i)(x + yi) into a real number. Sounds intriguing, right? Let's break it down step by step.

Understanding Complex Numbers and Real Numbers

Before we jump into solving the problem, let's quickly recap what complex and real numbers are. A real number is any number that can be plotted on a number line – think of integers, fractions, and decimals. Now, a complex number is a bit more interesting. It has two parts: a real part and an imaginary part. It's typically written in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit, defined as the square root of -1. So, when we multiply complex numbers, we need to remember how i behaves, especially that i^2 = -1. This property is key to simplifying complex expressions and figuring out when they become real. When we say an expression should represent a real number, we mean that the imaginary part of the result should be zero. This is because a real number, in the context of complex numbers, is simply a complex number with an imaginary part of 0, like 5 + 0i, which is just 5. So, our goal is to manipulate the given expression and find the values of x and y that make the imaginary component vanish. This involves a bit of algebra and careful handling of the imaginary unit, i. To master complex numbers, you need to understand their structure and how they interact under basic operations like addition, subtraction, multiplication, and division. Each operation follows specific rules that stem from the fundamental property of i. So, let's put on our mathematical hats and get to work!

Expanding the Expression

Okay, let's get our hands dirty with some algebra. Our mission is to expand the expression (4 + 5i)(x + yi). We’ll use the FOIL method (First, Outer, Inner, Last) to make sure we multiply everything correctly. This means we'll multiply the first terms, then the outer terms, then the inner terms, and finally the last terms. When multiplying complex numbers, every term in the first complex number must be multiplied by every term in the second complex number. So, let's break it down:

  • First: 4 * x = 4x
  • Outer: 4 * yi = 4yi
  • Inner: 5i * x = 5xi
  • Last: 5i * yi = 5yi^2

Now, let's put it all together. Our expanded expression looks like this: 4x + 4yi + 5xi + 5yi^2. But wait, we're not done yet! We need to simplify this expression, especially the 5yi^2 term. Remember that i^2 is equal to -1, so we can substitute that in. Doing so changes 5yi^2 to 5y(-1), which simplifies to -5y. Now our expression looks cleaner and more manageable. It's crucial to remember this basic property of i because it's what allows us to move between the complex and real domains. Misunderstanding or forgetting this can lead to significant errors in calculations involving complex numbers. With this simplification, we're one step closer to isolating the real and imaginary parts of our complex expression, which is essential for solving the problem at hand. So, keep that in mind as we proceed to the next step of grouping and setting up our equations.

Grouping Real and Imaginary Terms

Alright, we've expanded the expression, and now it's time to group the real and imaginary parts. This is a crucial step because it helps us clearly see the two components of our complex number. Remember, we want the imaginary part to be zero for the whole expression to be a real number. From our expanded and simplified expression 4x + 4yi + 5xi - 5y, let’s identify the terms without i (the real parts) and the terms with i (the imaginary parts). The real parts are 4x and -5y, while the imaginary parts are 4yi and 5xi. Now, we can rewrite the expression by grouping these terms together. We'll put the real parts first and then the imaginary parts. This gives us (4x - 5y) + (4y + 5x)i. Notice how we've factored out i from the imaginary terms to make it clear what the imaginary component is. This grouped form is super helpful because it visually separates the real and imaginary parts, making it easier to set up the equations we need to solve. This is a standard technique when dealing with complex numbers, especially when solving for variables that affect both real and imaginary components. By isolating the imaginary part, we can set it equal to zero and solve for the conditions under which the entire expression becomes real. This step is not just about organization; it's about setting up a pathway to the solution. With this clear separation, we're ready to move on to the next step: setting the imaginary part to zero and solving for our variables.

Setting the Imaginary Part to Zero

Now for the pivotal moment! We know that for the expression to represent a real number, the imaginary part must equal zero. We've neatly grouped our expression into real and imaginary parts: (4x - 5y) + (4y + 5x)i. So, our mission now is to make the imaginary part, which is (4y + 5x), equal to zero. This gives us the equation 4y + 5x = 0. This equation is the key to unlocking the solution. It tells us the relationship that x and y must satisfy for the original expression to be real. Think of it as a constraint – x and y can't just be any numbers; they have to play by the rules of this equation. Now, we have a single equation with two variables, which means we can't find unique values for x and y directly. Instead, we'll find a relationship between them. We can express y in terms of x (or vice versa) to understand this relationship better. This is a common technique in algebra when you have fewer equations than variables. By expressing one variable in terms of the other, we can see how they depend on each other to satisfy the condition we've set. This step is critical because it transforms our problem from finding specific values to understanding a general condition. Now, let's rearrange the equation to isolate one of the variables and see what that relationship looks like.

Solving for the Relationship Between x and y

Let's solve our equation, 4y + 5x = 0, to find the relationship between x and y. We can start by isolating y. Subtract 5x from both sides of the equation to get 4y = -5x. Now, divide both sides by 4 to solve for y: y = (-5/4)x. Ta-da! We've found the relationship. This equation tells us that y must be -5/4 times x for the original expression to be a real number. This is a powerful result because it gives us a whole set of solutions, not just one. For any value we choose for x, we can find a corresponding value for y that makes the expression real. This is a classic example of how algebra can reveal underlying structures and relationships between variables. It also shows how a single condition (the imaginary part being zero) can lead to a broad set of solutions. Instead of pinpointing specific numbers, we've identified a rule that governs the connection between x and y. This is a much more versatile outcome because it allows us to generate infinitely many solutions. So, now that we have this relationship, let's see how it applies to the answer choices we're given and find the correct one.

Checking the Answer Choices

Now that we have the relationship y = (-5/4)x, let's check the answer choices to see which pair of x and y values satisfies this condition. This is a crucial step to make sure we haven't made any mistakes and to pinpoint the correct answer. Remember, the correct answer will be the one where the y value is -5/4 times the x value.

Let's go through the options:

  • A. x = 4, y = 5: Plug these values into our equation: 5 = (-5/4) * 4. This simplifies to 5 = -5, which is not true. So, option A is incorrect.
  • B. x = -4, y = 0: Substitute these values: 0 = (-5/4) * -4. This simplifies to 0 = 5, which is also not true. Option B is incorrect.
  • C. x = 4, y = -5: Let's try these: -5 = (-5/4) * 4. This simplifies to -5 = -5, which is true! Option C satisfies our condition.
  • D. x = 0, y = 5: Finally, let's check this: 5 = (-5/4) * 0. This simplifies to 5 = 0, which is not true. Option D is incorrect.

It looks like we have a winner! Option C, x = 4 and y = -5, is the only pair that satisfies the relationship we found. This confirms that our algebraic manipulations and reasoning were correct. Checking the answer choices is always a good practice, as it helps catch any potential errors and ensures that the solution aligns with the specific constraints of the problem. So, with confidence, we can say that the correct answer is option C.

Conclusion

Alright, guys, we did it! We successfully navigated the world of complex numbers and found the values of x and y that make the expression (4 + 5i)(x + yi) a real number. By expanding the expression, grouping the real and imaginary parts, setting the imaginary part to zero, and solving for the relationship between x and y, we were able to find the condition y = (-5/4)x. Then, by checking the answer choices, we confirmed that x = 4 and y = -5 is the correct solution. This problem is a great example of how algebra and complex number theory come together. It highlights the importance of understanding the properties of complex numbers, especially the imaginary unit i, and how to manipulate them algebraically. Remember, the key to solving problems like this is to break them down into smaller, manageable steps. Each step, from expanding the expression to checking the answer choices, builds upon the previous one to lead us to the final solution. So, keep practicing, and you'll become a pro at complex numbers in no time! Keep an eye out for more math adventures here at Plastik Magazine, and until next time, keep those numbers crunching!