Problem 35 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 35-38: Find the values of \(x\) and \(y\), where \(x\) and \(y\) are real numbers. $$ 4+(x+2 y) i=x+2 i $$

Step-by-Step Solution

Answer

\( x = 4 \), \( y = -1 \)

Step 1: Separate Real and Imaginary Parts

Given the equation \( 4 + (x + 2y)i = x + 2i \), we begin by grouping the real parts and the imaginary parts on both sides of the equation. This gives us the system: Real part: \( 4 = x \) Imaginary part: \( x + 2y = 2 \).

Step 2: Solve for x

From the real part equation, \( 4 = x \), we can immediately find that \( x = 4 \).

Step 3: Substitute x into Imaginary Part Equation

Now that we have found \( x = 4 \), substitute this value into the equation from the imaginary part: \( 4 + 2y = 2 \).

Step 4: Solve for y

With the equation \( 4 + 2y = 2 \), solve for \( y \): Subtract 4 from both sides to get \( 2y = -2 \). Divide both sides by 2 to isolate \( y \): \( y = -1 \).

Key Concepts

Understanding Real Numbers

Real numbers encompass all the numbers we typically use in everyday arithmetic. These include both positive and negative numbers, as well as zero. Real numbers can be whole numbers, also known as integers, or they can be fractions or decimals, also termed rational numbers. Let’s break it down further:

• Integers: Whole numbers that can be positive, negative, or zero (e.g., -3, 0, 4).
• Rational Numbers: Numbers that can be expressed as a fraction of two integers, like 1/2 or 0.75.
• Irrational Numbers: These numbers cannot be written as a simple fraction, such as the square root of 2 or π.

Real numbers are fundamental in mathematics because they provide a complete set of numbers for arithmetic and algebraic operations. We often use them to solve equations where real solutions are necessary to make sense in practical scenarios. In the exercise, both variables, \(x\) and \(y\), are real numbers. This implies that their solutions must fit within the continuum of values that real numbers provide. By solving the real part of the equation, we directly find that \(x = 4\), which is a real number and adheres to the conditions given.

Exploring Imaginary Numbers

Imaginary numbers are a fascinating extension of the number system. They introduce the concept of the square root of negative numbers, which real numbers cannot address. The basic unit of imaginary numbers is \(i\), defined by the property \(i^2 = -1\). Here are some key points about imaginary numbers:

• They help solve equations that don't have real solutions, such as \(x^2 + 1 = 0\).
• Imaginary numbers combined with real numbers form complex numbers, expressed as \(a + bi\), where \(a\) and \(b\) are real numbers.
• In the equation from the exercise, \(i\) helps separate the real and imaginary components, simplifying the process of solving for \(x\) and \(y\).

Through the context of the problem, the imaginary part equation \(x + 2y = 2\) lets us separate the components effectively. Solving such equations often involves manipulating the imaginary part to find real solutions pertinent to the real-world problems or further mathematical contexts.

Solving Algebraic Equations

Algebraic equations are integral to mathematics, serving as the basis for solving problems that entail finding unknown values. These equations can take various forms, such as linear, quadratic, or polynomial, and the techniques to solve them will depend on the equation's complexity. Let's breakdown the essentials of solving algebraic equations:

• Identify the type of equation based on its structure.
• Use algebraic methods, such as substitution or elimination, to isolate variables.
• Apply arithmetic operations carefully to simplify equations.

In this exercise, we tackled a linear system derived from a complex number equation. By using the substitution method, once we had \(x = 4\) from the real component equation, we plugged this value into the imaginary component equation \(4 + 2y = 2\) to find \(y = -1\). This systematic approach ensures precise results by maintaining the equation's balance and accurately interpreting each component of the complex equation.