Question

Solve for all possible values of the real numbersand in the following equations.

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

Short answer

The answers are obtained:

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

Step-by-step solution

Given information

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

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:

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

Assume$x+iy=Z$and continue simplification:

$$\begin{gathered} \frac{Z+2+3i}{2Z-3}=2+i \\ \left(2+i\right)\left(2Z-3\right)=\left(Z+2+3i\right) \\ 4Z+2iZ-6-3i=Z+2+3i \\ 4Z+2iZ-6-3i=Z+2+3i \end{gathered}$$

Continue simplification:

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

Equate like terms

Equate like terms from both sides:

$$\begin{gathered} x+iy=\frac{36+2i}{13} \\ x=\frac{36}{13} \\ and \\ y=\frac{2}{13} \end{gathered}$$

Thus, the answers are obtained:

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