Q46P - Mathematical Methods in Physical Sciences

Question

Solve for all possible values of the real numbers x and y in the following equations $\frac{\mathbf{x}\mathbf{+}\mathbf{i}\mathbf{y}}{\mathbf{x}\mathbf{-}\mathbf{i}\mathbf{y}}\mathbf{=}\mathbf{-}\mathit{i}$.

Step-by-Step Solution

Answer

The answer is $x=-y$. 

Step 1: Given information

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

Step 2: Definition of a complex number

Every complex number can be described as: 

 

 z=a+bi

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

Step 3: Begin by stating the given information

State the given information:

 $$\begin{gathered} \frac{x+iy}{x-iy}=-i \\ x+iy=-i\left(x-iy\right) \\ x+iy=-ix-iy \end{gathered}$$

Step 4: Equate like terms

Equate like terms from both sides:

$$\begin{gathered} \left(x+y\right)+\left(x+y\right)i=0+0i \\ x+y=0 \\ x+y=0 \\ x=-y \end{gathered}$$ 

 Thus, the answer is $x=-y.$