Problem 38 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 35-38: Find the values of \(x\) and \(y\), where \(x\) and \(y\) are real numbers. $$ 8+(3 x+y) i=2 x-4 i $$

Step-by-Step Solution

Answer

The solution is \(x = 4\) and \(y = -16\).

Step 1: Identify the real and imaginary components on both sides

We start by writing the given complex equation:\[8 + (3x + y)i = 2x - 4i\]In a complex number of the form \(a + bi\), \(a\) is the real part and \(b\) is the imaginary part. So, we have the real parts as \(8\) and \(2x\) respectively, and the imaginary parts as \((3x + y)i\) and \(-4i\).

Step 2: Set up equations from the corresponding parts

Since the real and imaginary parts are equal, we set up the following equations:1. Real part: \[ 8 = 2x \]2. Imaginary part: \[ 3x + y = -4 \]

Step 3: Solve the equation for the real part

From the equation \(8 = 2x\), divide both sides by 2 to solve for \(x\):\[x = \frac{8}{2} = 4\]

Step 4: Substitute the value of x into the imaginary part equation

Substitute \(x = 4\) into the equation for the imaginary part:\[3(4) + y = -4\]which simplifies to:\[ 12 + y = -4 \]

Step 5: Solve the equation for y

Subtract 12 from both sides to solve for \(y\):\[y = -4 - 12 = -16\]

Key Concepts

Understanding the Real Part

In the world of complex numbers, the real part is an integral value representing the non-imaginary component of a number. Complex numbers are expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) represents the coefficient of the imaginary part.

In our given equation, \(8 + (3x + y)i = 2x - 4i\), the real part on the left side is \(8\) and on the right side is \(2x\).

The equation for the real part equates these:

• 8 = 2x

By solving this equation, you can isolate and determine the real number \(x\). This is the beauty of separating the real parts—the solution becomes straightforward.

Discovering the Imaginary Part

The imaginary part of a complex number captures the coefficient of \(i\), an imaginary unit satisfying \(i^2 = -1\). When working with complex numbers, the term multiplying \(i\) is essential in defining the number's imaginary component.

For the complex equation \(8 + (3x + y)i = 2x - 4i\), the imaginary part is given by the expressions:

• \((3x + y)\) on the left, corresponding to \(\text{i}\)
• \(-4\) on the right, also corresponding to \(\text{i}\)

The equation for these imaginary parts is derived as follows:

• \(3x + y = -4\)

This equation offers insight into the values of \(x\) and \(y\) when solved in conjunction with the real part equation.

Formulating Equations

Creating equations from the components of a complex number allows us to solve problems step by step. The process involves separating each component into its own concise mathematical expression.

By breaking down our complex number \(8 + (3x + y)i = 2x - 4i\) into real and imaginary parts, two main equations emerge:

• Real: \(8 = 2x\)
• Imaginary: \(3x + y = -4\)

These equations function independently based on their specific component properties, enabling smoother computation and solution finding for the unknown variables.

Arriving at the Solutions

Solving the equations obtained from complex numbers is a process that leads to determining the variables involved. For our example, once we have the two equations—one from the real part (\(8 = 2x\)) and one from the imaginary part (\(3x + y = -4\))—we can easily solve them to find \(x\) and \(y\).

Here's how it's done:

• Solve \(8 = 2x\) to get \(x = 4\).
• Substitute \(x = 4\) into the other equation: \(3(4) + y = -4\). Calculate \(12 + y = -4\).
• Finally, solve for \(y\) to find that \(y = -16\).

This structured approach to finding solutions showcases the efficacy of separating real and imaginary parts in complex number equations.