Chapter 1: Q48P (page 1)

Solve for all possible values of the real numbersand in the following equations.
$\frac{x + i y + 2 + 3 i}{2 x + 2 i y - 3} = i + 2$

Short Answer

Expert verified

The answers are obtained:

$x = \frac{36}{13}, y = \frac{2}{13}$

Step by step solution

Given information

It is given that $\frac{x + i y + 2 + 3 i}{2 x + 2 i y - 3} = i + 2 .$ .

Definition of a complex number

Every complex number can be represented as:

z=a+bi

Whereaandbare both real numbers, z is the complex number, and i is known as iota, which makes z a complex number.

Step 3: State the given information and simplify.

State the given information:

$\frac{x + i y + 2 + 3 i}{2 x + 2 i y - 3} = i + 2$

Assume $x + i y = Z$ and continue simplification:

$\frac{Z + 2 + 3 i}{2 Z - 3} = 2 + i (2 + i) (2 Z - 3) = (Z + 2 + 3 i) 4 Z + 2 i Z - 6 - 3 i = Z + 2 + 3 i 4 Z + 2 i Z - 6 - 3 i = Z + 2 + 3 i$

Continue simplification:

$3 Z + 2 i Z = 8 + 6 i Z = \frac{8 + 6 i}{3 + 2 i} Z = \frac{8 + 6 i}{3 + 2 i} \times \frac{3 - 2 i}{3 - 2 i} Z = \frac{24 + 12 - 16 i + 18 i}{3^{2} + 2^{2}}$

Equate like terms

Equate like terms from both sides:

$x + i y = \frac{36 + 2 i}{13} x = \frac{36}{13} a n d y = \frac{2}{13}$

Thus, the answers are obtained:

$x = \frac{36}{13}, y = \frac{2}{13}$

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Solve for all possible values of the real numbersand in the following equations.
$\frac{x + i y + 2 + 3 i}{2 x + 2 i y - 3} = i + 2$

Short Answer

Step by step solution

Given information

Definition of a complex number

Step 3: State the given information and simplify.

Equate like terms

One App. One Place for Learning.

Most popular questions from this chapter

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Solve for all possible values of the real numbersand in the following equations.x+iy+2+3i2x+2iy-3=i+2

Short Answer

Step by step solution

Given information

Definition of a complex number

Step 3: State the given information and simplify.

Equate like terms

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Geometrical and Physical Optics

Mechanics and Materials

Scientific Method Physics

Particle Model of Matter

Waves Physics

Energy Physics

Study anywhere. Anytime. Across all devices.

Solve for all possible values of the real numbersand in the following equations.
$\frac{x + i y + 2 + 3 i}{2 x + 2 i y - 3} = i + 2$