Solve For { X $}$ Given The Equation:${ (x + Yi)(3 + 4i) = 3 - 4i, \quad X \in \Re }$

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Introduction

In mathematics, solving equations involving complex numbers is a crucial aspect of algebra and analysis. The given equation, $(x + yi)(3 + 4i) = 3 - 4i, \quad x \in \Re$, is a complex equation that requires careful manipulation to isolate the variable $x$. In this article, we will delve into the world of complex numbers and explore the steps necessary to solve for $x$.

Understanding Complex Numbers

Before we dive into the solution, it's essential to understand 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 $i^2 = -1$. The real part of a complex number is $a$, and the imaginary part is $b$. In the given equation, $x$ and $y$ are real numbers, and $i$ is the imaginary unit.

Expanding the Equation

To solve for $x$, we need to expand the left-hand side of the equation using the distributive property. This will allow us to simplify the equation and isolate the variable $x$.

$(x + yi)(3 + 4i) = x(3 + 4i) + yi(3 + 4i)$

Using the distributive property, we can expand the equation further:

$x(3 + 4i) + yi(3 + 4i) = 3x + 4xi + 3yi + 4yi^2$

Since $i^2 = -1$, we can substitute this value into the equation:

$3x + 4xi + 3yi + 4yi^2 = 3x + 4xi + 3yi - 4y$

Simplifying the Equation

Now that we have expanded the equation, we can simplify it by combining like terms. We can group the real parts and imaginary parts separately:

$(3x - 4y) + (4x + 3y)i = 3 - 4i$

Equating Real and Imaginary Parts

Since the equation is an equality, we can equate the real parts and imaginary parts separately. This will give us two equations:

$3x - 4y = 3$ $4x + 3y = -4$

Solving the System of Equations

We now have a system of two equations with two variables. We can solve this system using substitution or elimination. Let's use substitution to solve for $x$.

Rearranging the first equation, we get:

$3x = 3 + 4y$

Dividing both sides by 3, we get:

$x = 1 + \frac{4}{3}y$

Substituting into the Second Equation

Now that we have expressed $x$ in terms of $y$, we can substitute this expression into the second equation:

$4(1 + \frac{4}{3}y) + 3y = -4$

Expanding and simplifying the equation, we get:

$4 + \frac{16}{3}y + 3y = -4$

Combining like terms, we get:

$\frac{16}{3}y + 3y = -8$

Solving for $y$

Now that we have a single equation with a single variable, we can solve for $y$. Let's multiply both sides by 3 to eliminate the fraction:

$16y + 9y = -24$

Combining like terms, we get:

$25y = -24$

Dividing both sides by 25, we get:

$y = -\frac{24}{25}$

Substituting $y$ into the Expression for $x$

Now that we have found the value of $y$, we can substitute this value into the expression for $x$:

$x = 1 + \frac{4}{3}(-\frac{24}{25})$

Simplifying the expression, we get:

$x = 1 - \frac{32}{25}$

$x = -\frac{3}{25}$

Conclusion

In this article, we have solved the equation $(x + yi)(3 + 4i) = 3 - 4i, \quad x \in \Re$ for the variable $x$. We have used complex number arithmetic and algebraic manipulation to isolate the variable $x$. The final solution is $x = -\frac{3}{25}$. This result demonstrates the importance of understanding complex numbers and their applications in mathematics.

Applications of Complex Numbers

Complex numbers have numerous applications in mathematics, physics, and engineering. Some of the key applications include:

• Signal Processing: Complex numbers are used to represent signals in the frequency domain, allowing for efficient filtering and analysis of signals.
• Electrical Engineering: Complex numbers are used to analyze and design electrical circuits, including filters, amplifiers, and oscillators.
• Physics: Complex numbers are used to describe the behavior of physical systems, including quantum mechanics and electromagnetism.
• Computer Science: Complex numbers are used in algorithms for solving linear systems, finding eigenvalues, and performing numerical computations.

Future Directions

The study of complex numbers is an active area of research, with many open problems and applications waiting to be explored. Some of the key areas of research include:

• Complex Analysis: The study of complex functions and their properties, including analytic continuation, singularities, and residues.
• Number Theory: The study of properties of complex numbers, including algebraic and transcendental numbers.
• Algebraic Geometry: The study of geometric objects defined by polynomial equations, including complex algebraic curves and surfaces.

References

• "Complex Analysis" by Serge Lang
• "Algebraic Geometry" by Robin Hartshorne
• "Number Theory" by George E. Andrews

Further Reading

For further reading on complex numbers and their applications, we recommend the following resources:

• "Complex Numbers and the Complex Plane" by Paul J. Nahin
• "The Complex Numbers" by John Stillwell
• "Complex Analysis: A First Course" by Serge Lang

By understanding complex numbers and their applications, we can unlock new insights and solutions to problems in mathematics, physics, and engineering.

Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in physics, engineering, and computer science. In this article, we will answer some of the most frequently asked questions about complex numbers.

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 $i^2 = -1$.

Q: What is the imaginary unit $i$?

A: The imaginary unit $i$ is a mathematical concept that is used to extend the real numbers to the complex numbers. It is defined as the square root of $-1$, and it satisfies the equation $i^2 = -1$.

Q: How do I add complex numbers?

A: To add complex numbers, you simply add the real parts and the imaginary parts separately. For example, if you have two complex numbers $a + bi$ and $c + di$, their sum is $(a + c) + (b + d)i$.

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you use the distributive property and the fact that $i^2 = -1$. For example, if you have two complex numbers $a + bi$ and $c + di$, their product is $(ac - bd) + (ad + bc)i$.

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 expressions and to eliminate the imaginary part.

Q: How do I find the magnitude of a complex number?

A: The magnitude of a complex number $a + bi$ is given by the formula $\sqrt{a^2 + b^2}$. The magnitude of a complex number is a measure of its size or length.

Q: What is the argument of a complex number?

A: The argument of a complex number $a + bi$ is the angle that the complex number makes with the positive real axis. It is given by the formula $\tan^{-1}\left(\frac{b}{a}\right)$.

Q: How do I convert a complex number to polar form?

A: To convert a complex number to polar form, you use the formula $r(\cos\theta + i\sin\theta)$, where $r$ is the magnitude of the complex number and $\theta$ is its argument.

Q: What is the difference between a complex number and a real 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. A real number is a number that can be expressed in the form $a$, where $a$ is a real number.

Q: Can complex numbers be used in real-world applications?

A: Yes, complex numbers have numerous applications in physics, engineering, and computer science. They are used to describe the behavior of electrical circuits, to analyze signals, and to solve problems in quantum mechanics.

Q: Are complex numbers used in everyday life?

A: Yes, complex numbers are used in many everyday applications, such as in the design of electrical circuits, in the analysis of signals, and in the solution of problems in physics and engineering.

Q: Can complex numbers be used to solve problems in mathematics?

A: Yes, complex numbers are used to solve problems in mathematics, such as in the solution of equations, in the analysis of functions, and in the study of geometry.

Q: Are complex numbers difficult to understand?

A: Complex numbers can be challenging to understand at first, but with practice and experience, they become easier to work with. It is essential to have a good understanding of the basics of complex numbers before moving on to more advanced topics.

Q: Can complex numbers be used to solve problems in other fields?

A: Yes, complex numbers have applications in many fields, including physics, engineering, computer science, and mathematics. They are used to describe the behavior of systems, to analyze signals, and to solve problems in many different areas.

Conclusion

In this article, we have answered some of the most frequently asked questions about complex numbers. We have covered topics such as the definition of complex numbers, the imaginary unit, addition and multiplication of complex numbers, conjugates, magnitudes, and arguments. We have also discussed the applications of complex numbers in physics, engineering, and computer science.