Solving Complex Equations: Find Xy

Hey math enthusiasts! Today, we're diving into a cool problem involving complex numbers. We've got an equation: $\frac{x}{1- i }+\frac{y}{1-2 i }=\frac{5}{1-3 i }$, and our mission, should we choose to accept it, is to find the value of $xy$. Sounds like fun, right? Let's break it down step-by-step. This is a classic example of how complex numbers can be manipulated and solved using algebraic techniques. We'll start by rationalizing the denominators and then separating the real and imaginary parts to form a system of equations. From there, it's all about solving for x and y, and finally, finding their product.

Understanding the Problem: Complex Numbers and Equations

Alright, let's get our heads in the game. We're dealing with complex numbers here. Remember those? They're numbers 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. So, in our equation, we see terms like $1 - i$, $1 - 2i$, and $1 - 3i$. These are all complex numbers. Our equation is an equation with complex numbers, and x and y are real numbers. The goal is to figure out the values of x and y that satisfy this equation. Then, we multiply them to find xy. This kind of problem often appears in math contests and is a good exercise in understanding complex number arithmetic. The core concept here is to manipulate the equation to isolate the real and imaginary parts. By doing so, we can convert a complex equation into a system of real equations, which we can then solve using standard algebraic methods. It's like a puzzle, and each step brings us closer to the solution.

Now, before we get our hands dirty with the calculations, let's quickly review the basics. Complex numbers are added and subtracted by combining their real and imaginary parts. Multiplication involves using the distributive property, remembering that $i^2 = -1$. Division is a bit trickier, as we need to rationalize the denominator by multiplying both the numerator and denominator by the complex conjugate. The complex conjugate of a complex number $a + bi$ is $a - bi$. This process eliminates the imaginary part from the denominator, making it a real number.

In our equation, the complex numbers are in the denominators. So, we'll need to rationalize those fractions. This will be the first step in simplifying the equation and separating the real and imaginary parts. Once we've done that, we'll have a much clearer picture of what x and y need to be. And finally, remember that in an equation, two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. This principle will be crucial in setting up our system of equations.

Rationalizing the Denominators: The First Step

Okay, guys, let's get down to business. Our first step is to get rid of those pesky complex numbers in the denominators. This process is called rationalizing the denominator, and it's super important in complex number arithmetic. To do this, we multiply both the numerator and denominator of each fraction by the complex conjugate of the denominator. Remember, the complex conjugate of $a + bi$ is $a - bi$. So let's start with $\frac{x}{1- i }$. The complex conjugate of $1 - i$ is $1 + i$. Multiplying the numerator and denominator by this, we get: $\frac{x(1 + i)}{(1 - i)(1 + i)}$.

Let's simplify that. The denominator becomes $(1 - i^2)$, which is $(1 - (-1))$, or $2$. So the first term simplifies to $\frac{x(1 + i)}{2}$. Next, let's look at the second term, $\frac{y}{1-2 i }$. The complex conjugate of $1 - 2i$ is $1 + 2i$. Multiplying the numerator and denominator by this, we get: $\frac{y(1 + 2 i)}{(1 - 2 i)(1 + 2 i)}$. The denominator simplifies to $(1 - (2i)^2)$, or $(1 - (-4))$, which is $5$. So the second term simplifies to $\frac{y(1 + 2 i)}{5}$. Finally, let's tackle the right side of the equation, $\frac{5}{1-3 i }$. The complex conjugate of $1 - 3i$ is $1 + 3i$. Multiplying the numerator and denominator by this, we get: $\frac{5(1 + 3 i)}{(1 - 3 i)(1 + 3 i)}$.

The denominator simplifies to $(1 - (3i)^2)$, or $(1 - (-9))$, which is $10$. Thus, the right side simplifies to $\frac{5(1 + 3 i)}{10}$, or $\frac{1 + 3i}{2}$. So, our equation now looks like this: $\frac{x(1 + i)}{2} + \frac{y(1 + 2 i)}{5} = \frac{1 + 3i}{2}$. We've successfully rationalized all the denominators, making it easier to separate the real and imaginary parts. This is a crucial step because it transforms our equation from one involving complex fractions to one that we can work with more directly.

This process is like cleaning up a messy equation. By rationalizing the denominators, we've removed the complex numbers from the bottom, making it easier to compare the real and imaginary parts. It's similar to simplifying fractions in regular algebra. Now that we've done this, the next step is to expand the terms and group the real and imaginary components together. This will set us up for solving for x and y. Remember to be careful with your calculations, and double-check your work to avoid any mistakes. This part can get a bit tedious, but it's essential for getting the correct answer.

Separating Real and Imaginary Parts: Setting Up Equations

Alright, buckle up, because now we're going to expand and separate the real and imaginary parts of our equation. It's like sorting ingredients before you start cooking. We have $\frac{x(1 + i)}{2} + \frac{y(1 + 2 i)}{5} = \frac{1 + 3i}{2}$. Let's expand the left side. The first term becomes $\frac{x}{2} + \frac{xi}{2}$, and the second term becomes $\frac{y}{5} + \frac{2yi}{5}$. Now, our equation looks like this: $\frac{x}{2} + \frac{xi}{2} + \frac{y}{5} + \frac{2yi}{5} = \frac{1 + 3i}{2}$. Let's rearrange this to group the real and imaginary parts. The real parts are $\frac{x}{2}$ and $\frac{y}{5}$, and the imaginary parts are $\frac{x}{2}i$ and $\frac{2y}{5}i$. On the right side, the real part is $\frac{1}{2}$ and the imaginary part is $\frac{3}{2}i$. So we can rewrite the equation as: $(\frac{x}{2} + \frac{y}{5}) + i(\frac{x}{2} + \frac{2y}{5}) = \frac{1}{2} + \frac{3}{2}i$.

Now, here's the key: for two complex numbers to be equal, their real parts must be equal, and their imaginary parts must be equal. Therefore, we can set up two separate equations. The first equation, based on the real parts, is $\frac{x}{2} + \frac{y}{5} = \frac{1}{2}$. The second equation, based on the imaginary parts, is $\frac{x}{2} + \frac{2y}{5} = \frac{3}{2}$. We now have a system of two linear equations with two variables, x and y. This is something we can solve using methods like substitution or elimination.

This separation is critical because it transforms a single complex equation into a pair of real equations that we can solve using familiar algebraic techniques. Think of it like taking a complex puzzle and breaking it down into smaller, more manageable pieces. By equating the real and imaginary parts, we create a clear path to find the values of x and y. Solving these equations is a fundamental skill in algebra, and it's directly applicable here. With the system of equations set up, the rest of the problem is straightforward algebra. Remember, always double-check your work, especially when dealing with multiple steps and equations. Also, keep in mind that the accuracy of your final answer depends on the correctness of each step along the way. Be careful with those fractions, and you'll do great!

Solving for x and y: The Algebraic Journey

Let's get those x and y values! We've got our system of equations: $\frac{x}{2} + \frac{y}{5} = \frac{1}{2}$ and $\frac{x}{2} + \frac{2y}{5} = \frac{3}{2}$. We can use either substitution or elimination to solve this. Let's use elimination. Notice that the $\frac{x}{2}$ term is the same in both equations. So, let's subtract the first equation from the second equation.

Subtracting $\frac{x}{2} + \frac{y}{5} = \frac{1}{2}$ from $\frac{x}{2} + \frac{2y}{5} = \frac{3}{2}$, we get: $(\frac{2y}{5} - \frac{y}{5}) = (\frac{3}{2} - \frac{1}{2})$. This simplifies to $\frac{y}{5} = 1$. Multiplying both sides by 5, we get $y = 5$.

Awesome, we have the value of y! Now, let's substitute this value back into one of the original equations to solve for x. Let's use the first equation: $\frac{x}{2} + \frac{y}{5} = \frac{1}{2}$. Substitute $y = 5$ into the equation: $\frac{x}{2} + \frac{5}{5} = \frac{1}{2}$. This simplifies to $\frac{x}{2} + 1 = \frac{1}{2}$. Subtracting 1 from both sides, we get $\frac{x}{2} = -\frac{1}{2}$. Multiplying both sides by 2, we find that $x = -1$.

There you have it! We've successfully solved for x and y. We found that $x = -1$ and $y = 5$. This process involved careful algebraic manipulation, and it's a testament to the power of breaking down a complex problem into smaller, solvable steps. Always double-check your solutions to ensure they satisfy the original equations, as this helps prevent errors. Now, let's use the values of $x$ and $y$ to find $xy$.

Finding xy: The Final Calculation

We're in the home stretch, guys! We've found that $x = -1$ and $y = 5$. Our goal is to find the value of $xy$. This is super easy now. We just multiply the values of x and y together: $xy = (-1)(5)$. Therefore, $xy = -5$.

And that's the answer! We started with a complex equation, rationalized denominators, separated real and imaginary parts, formed a system of equations, solved for x and y, and finally, calculated xy. It's a journey, but it's also a great way to reinforce our understanding of complex numbers and algebra. Finding $xy$ is the last step in the calculation. Always remember to check your work, especially in a problem that has multiple steps and calculations. It’s also good practice to make sure the final solution makes sense in the context of the problem.

Conclusion: Wrapping It Up

So, there you have it. We've successfully navigated the world of complex numbers and equations to find that $xy = -5$. This problem is a classic example of how to solve complex equations by rationalizing denominators, separating the real and imaginary parts, and using algebraic techniques to find the values of the variables. The key takeaways from this problem are the importance of understanding the basics of complex number arithmetic, the ability to manipulate equations, and the application of algebraic methods to solve for unknown variables. This type of problem is not just a mathematical exercise; it builds a strong foundation for more advanced concepts in mathematics and related fields.

Keep practicing, keep exploring, and remember that mathematics is all about problem-solving and critical thinking. Every problem you solve, every concept you understand, brings you closer to mastering the fascinating world of numbers and equations. Keep up the great work, and happy solving! If you enjoyed this explanation, you might also like to explore other topics in complex numbers, such as De Moivre's theorem or the complex plane. Happy learning! We hope you enjoyed this journey into the world of complex numbers. Remember to always double-check your work and to practice regularly. See you in the next one!