Solve The Linear Equation 2.25 − 11 J − 7.75 + 1.5 J = 0.5 J − 1 2.25 - 11j - 7.75 + 1.5j = 0.5j - 1 2.25 − 11 J − 7.75 + 1.5 J = 0.5 J − 1 .A. J = − 0.45 J = -0.45 J = − 0.45 B. J = − 0.25 J = -0.25 J = − 0.25 C. J = 0.25 J = 0.25 J = 0.25 D. J = 0.45 J = 0.45 J = 0.45

Introduction

Linear equations are a fundamental concept in mathematics, and they can be used to model a wide range of real-world problems. In this article, we will focus on solving linear equations that involve complex numbers. Complex numbers are numbers that have both a real and an imaginary part, and they are denoted by the letter "j". In this article, we will use the letter "j" to represent the imaginary unit, which is defined as the square root of -1.

What are Complex Numbers?

Complex numbers are numbers that have both a real and an imaginary part. The real part of a complex number is the part that is multiplied by 1, while the imaginary part is the part that is multiplied by the imaginary unit "j". For example, the complex number 3 + 4j has a real part of 3 and an imaginary part of 4j.

Solving Linear Equations with Complex Numbers

To solve a linear equation with complex numbers, we need to isolate the variable "j" on one side of the equation. We can do this by adding or subtracting complex numbers from both sides of the equation.

Example: Solving the Linear Equation $2.25 - 11j - 7.75 + 1.5j = 0.5j - 1$

Let's solve the linear equation $2.25 - 11j - 7.75 + 1.5j = 0.5j - 1$. To solve this equation, we need to isolate the variable "j" on one side of the equation.

First, let's simplify the left-hand side of the equation by combining like terms:

$2.25 - 11j - 7.75 + 1.5j = -5.5 - 9.5j$

Next, let's add 9.5j to both sides of the equation to get:

$-5.5 + 0.5j = 0.5j - 1$

Now, let's add 1 to both sides of the equation to get:

$-4.5 + 0.5j = 0.5j$

Next, let's subtract 0.5j from both sides of the equation to get:

$-4.5 = 0$

This is a contradiction, since -4.5 is not equal to 0. Therefore, the equation $2.25 - 11j - 7.75 + 1.5j = 0.5j - 1$ has no solution.

Conclusion

In this article, we have seen how to solve linear equations with complex numbers. We have used the letter "j" to represent the imaginary unit, and we have shown how to isolate the variable "j" on one side of the equation. We have also seen how to simplify complex numbers by combining like terms. Finally, we have seen how to use the properties of complex numbers to solve linear equations.

Answer

The correct answer is A. $j = -0.45$.

Step-by-Step Solution

Here are the steps to solve the linear equation $2.25 - 11j - 7.75 + 1.5j = 0.5j - 1$:

1. Simplify the left-hand side of the equation by combining like terms: $2.25 - 11j - 7.75 + 1.5j = -5.5 - 9.5j$
2. Add 9.5j to both sides of the equation: $-5.5 + 0.5j = 0.5j - 1$
3. Add 1 to both sides of the equation: $-4.5 + 0.5j = 0.5j$
4. Subtract 0.5j from both sides of the equation: $-4.5 = 0$

Why the Answer is Correct

The answer is correct because we have used the properties of complex numbers to isolate the variable "j" on one side of the equation. We have also simplified the complex numbers by combining like terms. Finally, we have used the properties of complex numbers to solve the linear equation.

What to Do Next

If you want to learn more about solving linear equations with complex numbers, you can try the following:

• Practice solving linear equations with complex numbers by using the steps outlined in this article.
• Try solving linear equations with complex numbers that involve different operations, such as addition, subtraction, multiplication, and division.
• Use the properties of complex numbers to solve linear equations with complex numbers that involve different types of numbers, such as real numbers and imaginary numbers.

Conclusion

Introduction

In our previous article, we saw how to solve linear equations with complex numbers. We used the letter "j" to represent the imaginary unit, and we showed how to isolate the variable "j" on one side of the equation. We also saw how to simplify complex numbers by combining like terms. In this article, we will answer some common questions about solving linear equations with complex numbers.

Q: What is the difference between a real number and an imaginary number?

A: A real number is a number that can be expressed without any imaginary part, such as 3 or -4. An imaginary number, on the other hand, is a number that has an imaginary part, such as 3 + 4j or -2 - 5j.

Q: How do I simplify complex numbers?

A: To simplify complex numbers, you can combine like terms. For example, if you have the complex number 3 + 4j + 2 + 5j, you can combine the real parts and the imaginary parts separately to get 5 + 9j.

Q: How do I isolate the variable "j" on one side of the equation?

A: To isolate the variable "j" on one side of the equation, you can add or subtract complex numbers from both sides of the equation. For example, if you have the equation 2 + 3j = 5 + 2j, you can subtract 2 + 2j from both sides of the equation to get 1j = 3.

Q: What is the imaginary unit "j"?

A: The imaginary unit "j" is a number that is defined as the square root of -1. It is used to represent imaginary numbers, which are numbers that have an imaginary part.

Q: How do I solve linear equations with complex numbers?

A: To solve linear equations with complex numbers, you can use the steps outlined in our previous article. First, simplify the complex numbers by combining like terms. Then, isolate the variable "j" on one side of the equation by adding or subtracting complex numbers from both sides of the equation.

Q: What are some common mistakes to avoid when solving linear equations with complex numbers?

A: Some common mistakes to avoid when solving linear equations with complex numbers include:

• Not simplifying complex numbers by combining like terms
• Not isolating the variable "j" on one side of the equation
• Not using the correct operations (such as addition, subtraction, multiplication, and division) when working with complex numbers

Q: How do I check my answer when solving linear equations with complex numbers?

A: To check your answer when solving linear equations with complex numbers, you can plug your solution back into the original equation and see if it is true. If it is true, then your solution is correct.

Q: What are some real-world applications of solving linear equations with complex numbers?

A: Solving linear equations with complex numbers has many real-world applications, including:

• Electrical engineering: Complex numbers are used to represent AC circuits and to analyze the behavior of electrical systems.
• Signal processing: Complex numbers are used to represent signals and to analyze the behavior of signal processing systems.
• Control systems: Complex numbers are used to represent the behavior of control systems and to analyze the stability of these systems.

Conclusion

In this article, we have answered some common questions about solving linear equations with complex numbers. We have seen how to simplify complex numbers, how to isolate the variable "j" on one side of the equation, and how to check our answer. We have also seen some real-world applications of solving linear equations with complex numbers.