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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

01

Given information

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

02

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.

03

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:

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

04

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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Most popular questions from this chapter

Use power series to evaluate the function at the given point. Compare with computer results, using the computer to find the series, and also to do the problem without series. Resolve any disagreement in results (see Example 1). $e^{sinx} - \frac{1}{x^{3}} In (1 + x^{3} e^{x}) at x = 0.00035$ .

$c o s (e^{x} - 1)$

$s i n [l n (1 + x)]$

$\frac{s i n (\sqrt{x})}{\sqrt{x}}, x > 0$

$\ln (1 + x e^{x})$

See all solutions

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