Solving Complex Equations (√2 X - Yi) * (1 + I) = 2-i

Hey guys! Let's dive into a fun math problem today. We're going to break down how to solve a complex equation step-by-step. This type of problem might seem a bit intimidating at first, but don't worry, we'll make it super clear and easy to understand. Our mission is to find the values of 'x' and 'y' in the equation (√2 x - yi) * (1 + i) = 2 - i. Ready? Let's get started!

Understanding the Complex Equation

At the heart of this problem is a complex equation, and understanding the equation is the first crucial step. Complex numbers are numbers that have both a real part and an imaginary part. They are usually written in the form a + bi, where 'a' is the real part, 'b' is the imaginary part, and 'i' is the imaginary unit, which is defined as the square root of -1 (i = √-1). In our equation, we have (√2 x - yi) * (1 + i) = 2 - i, where √2 x and -y are real numbers, and -yi and i are imaginary terms. The equation essentially states that when we multiply the complex number (√2 x - yi) by (1 + i), we get another complex number, 2 - i. To solve this, we need to use the rules of complex number arithmetic, particularly complex number multiplication, and then equate the real and imaginary parts on both sides of the equation.

To truly grasp the equation, it's essential to recognize the individual components and their roles. The term (√2 x - yi) is a complex number where √2 x is the real part and -yi is the imaginary part. Similarly, (1 + i) is another complex number with 1 as the real part and i as the imaginary part. The right side of the equation, 2 - i, is also a complex number with 2 as the real part and -1 as the coefficient of the imaginary unit i. Our goal is to find the values of x and y that make this equation true. This involves multiplying the complex numbers on the left side and then comparing the real and imaginary parts with those on the right side. By doing so, we create a system of equations that we can solve to find the values of x and y. It's like solving a puzzle where each part must fit perfectly to reveal the solution. Breaking down the equation into its components and understanding how they interact is the key to unlocking the solution.

Step-by-Step Solution

Now, let's jump into the step-by-step solution. The step-by-step solution will help you understand exactly how to tackle this kind of complex equation. First, we need to expand the left side of the equation using the distributive property (also known as the FOIL method): (√2 x - yi) * (1 + i). This means we multiply each term in the first complex number by each term in the second complex number. So, we have:

(√2 x * 1) + (√2 x * i) + (-yi * 1) + (-yi * i)

This simplifies to:

√2 x + √2 xi - yi - yi²

Remember that i² is equal to -1. So, we can replace yi² with -y(-1), which simplifies to +y. Our equation now looks like this:

√2 x + √2 xi - yi + y

Next, we need to group the real parts and the imaginary parts together. The real parts are √2 x and y, and the imaginary parts are √2 xi and -yi. So, we can rewrite the equation as:

(√2 x + y) + (√2 x - y)i

Now, we have a complex number in the form (a + bi), where 'a' is (√2 x + y) and 'b' is (√2 x - y). We know that this complex number is equal to 2 - i. This means that the real parts must be equal, and the imaginary parts must be equal. So, we can set up two equations:

√2 x + y = 2 (Equation 1) √2 x - y = -1 (Equation 2)

We now have a system of two equations with two variables, x and y. We can solve this system using various methods, such as substitution or elimination. Let's use the elimination method. We can add Equation 1 and Equation 2 together. Notice that the 'y' terms will cancel out:

(√2 x + y) + (√2 x - y) = 2 + (-1)

2√2 x = 1

Now, we can solve for x:

x = 1 / (2√2)

To rationalize the denominator, we multiply the numerator and denominator by √2:

x = (1 * √2) / (2√2 * √2) x = √2 / 4

Now that we have the value of x, we can substitute it back into either Equation 1 or Equation 2 to solve for y. Let's use Equation 1:

√2 * (√2 / 4) + y = 2

(2 / 4) + y = 2

1/2 + y = 2

y = 2 - 1/2 y = 3/2

So, we have found the values of x and y: x = √2 / 4 and y = 3/2. This is the solution to our complex equation!

Verifying the Solution

Alright, we've got our values for x and y, but let's not stop there! Verifying the solution is a critical step in problem-solving, especially with complex equations. It ensures we haven't made any sneaky errors along the way. To verify our solution, we'll substitute the values we found, x = √2 / 4 and y = 3/2, back into the original equation: (√2 x - yi) * (1 + i) = 2 - i.

Substituting the values, we get:

(√2 * (√2 / 4) - (3/2)i) * (1 + i)

Let's simplify this step by step. First, simplify the term inside the first parentheses:

(2/4 - (3/2)i) * (1 + i)

Which simplifies to:

(1/2 - (3/2)i) * (1 + i)

Now, we'll multiply these two complex numbers using the distributive property:

(1/2 * 1) + (1/2 * i) + (- (3/2)i * 1) + (- (3/2)i * i)

This simplifies to:

1/2 + (1/2)i - (3/2)i - (3/2)i²

Remember that i² is -1, so we can replace -(3/2)i² with +(3/2):

1/2 + (1/2)i - (3/2)i + 3/2

Now, let's group the real and imaginary parts:

(1/2 + 3/2) + ((1/2)i - (3/2)i)

This simplifies to:

(4/2) + (-2/2)i

Which further simplifies to:

2 - i

Tada! This matches the right side of our original equation, 2 - i. This means our solution, x = √2 / 4 and y = 3/2, is correct. We've successfully verified our solution, giving us confidence in our answer. Verifying not only confirms our solution but also deepens our understanding of the problem and the steps involved. It's like putting the final piece in a puzzle – satisfying and assuring!

Common Mistakes to Avoid

When dealing with complex equations, there are a few common mistakes to avoid that can trip you up. Recognizing these pitfalls can save you time and frustration. One of the most frequent errors is messing up the arithmetic when multiplying complex numbers. Remember to carefully apply the distributive property and to correctly handle the imaginary unit 'i'. Specifically, remember that i² = -1. Forgetting this simple fact can lead to incorrect signs and ultimately a wrong solution. Another common mistake is failing to group the real and imaginary terms correctly after expanding the equation. Make sure to clearly separate the real parts from the imaginary parts before setting up your system of equations. This step is crucial for accurately equating the real and imaginary parts on both sides of the equation.

Also, errors can creep in when solving the system of equations. Whether you're using substitution or elimination, be meticulous with your algebraic manipulations. A small mistake in one step can snowball into a wrong answer. Double-check your work, especially when dealing with fractions or square roots. Finally, don't skip the verification step! It's tempting to think you're done once you've found values for x and y, but substituting those values back into the original equation is essential to catch any errors. It's like proofreading a paper – it's the last line of defense against mistakes. By being aware of these common pitfalls and taking the time to double-check your work, you can confidently tackle complex equations and arrive at the correct solution. Remember, practice makes perfect, so keep at it, and you'll become a pro at solving these types of problems!

Tips and Tricks for Solving Complex Equations

To become a real whiz at solving complex equations, let's talk about some tips and tricks. These strategies can make the process smoother and more efficient. First off, always simplify your equation as much as possible before diving into calculations. Look for opportunities to combine like terms or reduce fractions. A simpler equation is always easier to work with. Next, when multiplying complex numbers, use the distributive property (FOIL method) carefully. Pay close attention to signs and remember that i² = -1. It's a small detail, but it makes a big difference. Another helpful trick is to rewrite complex numbers in their standard form (a + bi) whenever possible. This makes it easier to identify the real and imaginary parts and to perform operations like addition, subtraction, multiplication, and division.

When you end up with a system of equations, think strategically about how to solve it. Sometimes substitution is the best approach, while other times elimination is more efficient. Look for ways to simplify the equations before you start solving, such as multiplying both sides by a common denominator to eliminate fractions. Also, practice recognizing patterns in complex equations. Some equations might lend themselves to specific techniques, like using the complex conjugate. The complex conjugate of a complex number a + bi is a - bi. Multiplying a complex number by its conjugate results in a real number, which can be very useful for simplifying expressions. Finally, remember to always verify your solution by substituting it back into the original equation. This not only confirms your answer but also helps you catch any errors you might have made along the way. With these tips and tricks in your toolbox, you'll be well-equipped to tackle even the trickiest complex equations. So, keep practicing, and you'll become a complex number master in no time!

Conclusion

So, there you have it! We've journeyed through solving the complex equation (√2 x - yi) * (1 + i) = 2 - i, and hopefully, you found it super helpful. We started by understanding the basics of complex numbers, then we broke down the equation step-by-step, verified our solution, and even looked at common mistakes to avoid and some handy tips and tricks. Remember, the key to mastering complex equations is practice, practice, practice! The more you work with these types of problems, the more comfortable and confident you'll become. Don't be afraid to make mistakes – they're just learning opportunities in disguise. Keep these strategies in mind, and you'll be solving complex equations like a pro in no time. Keep up the great work, and happy problem-solving, guys!