What Are The Values Of X X X And Y Y Y That Satisfy This Equation? ( X + Y I ) + ( 4 + 9 I ) = 9 − 4 I (x + Yi) + (4 + 9i) = 9 - 4i ( X + Y I ) + ( 4 + 9 I ) = 9 − 4 I A. X = 5 X = 5 X = 5 And Y = 13 Y = 13 Y = 13 B. X = 5 X = 5 X = 5 And Y = − 13 Y = -13 Y = − 13 C. X = 9 X = 9 X = 9 And Y = − 4 Y = -4 Y = − 4 D.

Introduction

Complex equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic manipulation and complex number theory. In this article, we will explore the solution to a complex equation of the form $(x + yi) + (4 + 9i) = 9 - 4i$, where $x$ and $y$ are real numbers. We will break down the solution step by step, using a combination of algebraic manipulation and complex number properties.

Understanding Complex Numbers

Before we dive into the solution, let's take a moment to review the basics of complex numbers. 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, which satisfies the equation $i^2 = -1$. The real part of a complex number is $a$, and the imaginary part is $b$. Complex numbers can be added, subtracted, multiplied, and divided using the standard rules of arithmetic, with the additional rule that $i^2 = -1$.

Breaking Down the Equation

Now that we have a basic understanding of complex numbers, let's break down the given equation:

$(x + yi) + (4 + 9i) = 9 - 4i$

Our goal is to find the values of $x$ and $y$ that satisfy this equation. To do this, we will use a combination of algebraic manipulation and complex number properties.

Step 1: Combine Like Terms

The first step in solving this equation is to combine like terms. We can do this by adding the real parts and the imaginary parts separately:

$(x + 4) + (y + 9)i = 9 - 4i$

Step 2: Equate Real and Imaginary Parts

Now that we have combined like terms, we can equate the real and imaginary parts separately. This gives us two equations:

$x + 4 = 9$ ... (Equation 1) $y + 9 = -4$ ... (Equation 2)

Step 3: Solve for x and y

Now that we have two equations, we can solve for $x$ and $y$ separately. Let's start with Equation 1:

$x + 4 = 9$

Subtracting 4 from both sides gives us:

$x = 5$

Now that we have found the value of $x$, let's move on to Equation 2:

$y + 9 = -4$

Subtracting 9 from both sides gives us:

$y = -13$

Conclusion

In this article, we have solved the complex equation $(x + yi) + (4 + 9i) = 9 - 4i$ using a combination of algebraic manipulation and complex number properties. We have found that the values of $x$ and $y$ that satisfy this equation are $x = 5$ and $y = -13$. This solution demonstrates the importance of understanding complex numbers and their properties in solving mathematical equations.

Answer

The correct answer is:

A. $x = 5$ and $y = -13$

Discussion

This problem requires a deep understanding of complex numbers and their properties. The solution involves a combination of algebraic manipulation and complex number properties, making it a challenging problem for students. However, with practice and patience, students can develop the skills and confidence needed to solve complex equations like this one.

Additional Resources

For students who want to learn more about complex numbers and their properties, here are some additional resources:

• Khan Academy: Complex Numbers
• MIT OpenCourseWare: Complex Analysis
• Wolfram MathWorld: Complex Numbers

Introduction

In our previous article, we solved the complex equation $(x + yi) + (4 + 9i) = 9 - 4i$ using a combination of algebraic manipulation and complex number properties. In this article, we will answer some frequently asked questions about complex equations and provide additional resources for students who want to learn more.

Q: What is a complex number?

A: 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, which satisfies the equation $i^2 = -1$.

Q: How do I add complex numbers?

A: To add complex numbers, you can add the real parts and the imaginary parts separately. For example, if you have two complex numbers $a + bi$ and $c + di$, you can add them as follows:

$(a + bi) + (c + di) = (a + c) + (b + d)i$

Q: How do I subtract complex numbers?

A: To subtract complex numbers, you can subtract the real parts and the imaginary parts separately. For example, if you have two complex numbers $a + bi$ and $c + di$, you can subtract them as follows:

$(a + bi) - (c + di) = (a - c) + (b - d)i$

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you can use the distributive property and the fact that $i^2 = -1$. For example, if you have two complex numbers $a + bi$ and $c + di$, you can multiply them as follows:

$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

Q: How do I divide complex numbers?

A: To divide complex numbers, you can use the fact that $i^2 = -1$ and the conjugate of a complex number. For example, if you have two complex numbers $a + bi$ and $c + di$, you can divide them as follows:

$\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number $a + bi$ is $a - bi$. The conjugate of a complex number is used to simplify complex expressions and to divide complex numbers.

Q: How do I simplify complex expressions?

A: To simplify complex expressions, you can use the fact that $i^2 = -1$ and the conjugate of a complex number. For example, if you have a complex expression $a + bi + c + di$, you can simplify it as follows:

$a + bi + c + di = (a + c) + (b + d)i$

Conclusion

In this article, we have answered some frequently asked questions about complex equations and provided additional resources for students who want to learn more. We hope that this article has been helpful in clarifying some of the concepts and techniques involved in solving complex equations.

Additional Resources

For students who want to learn more about complex numbers and their properties, here are some additional resources:

• Khan Academy: Complex Numbers
• MIT OpenCourseWare: Complex Analysis
• Wolfram MathWorld: Complex Numbers

By following these resources and practicing complex equations, students can develop a deep understanding of complex numbers and their properties, and become proficient in solving mathematical equations.

Practice Problems

Here are some practice problems to help students develop their skills in solving complex equations:

1. Solve the complex equation $(x + yi) + (3 + 4i) = 5 - 2i$.
2. Solve the complex equation $(x + yi) - (2 + 3i) = 1 + 4i$.
3. Solve the complex equation $(x + yi)(2 + 3i) = 5 - 2i$.
4. Solve the complex equation $\frac{x + yi}{2 + 3i} = 1 + 4i$.

By practicing these problems and following the resources provided, students can develop a deep understanding of complex numbers and their properties, and become proficient in solving mathematical equations.