# If (4+i)/(2-3i) = x+iy for real numbers x, y, then what are x and y ?

Mar 17, 2017

$x = \frac{5}{13} \text{ }$ and $\text{ } y = \frac{14}{13}$

#### Explanation:

$x + i y = \frac{4 + i}{2 - 3 i}$

$\textcolor{w h i t e}{x + i y} = \frac{\left(4 + i\right) \left(2 + 3 i\right)}{\left(2 - 3 i\right) \left(2 + 3 i\right)}$

$\textcolor{w h i t e}{x + i y} = \frac{8 + 12 i + 2 i + 3 {i}^{2}}{{2}^{2} - {\left(3 i\right)}^{2}}$

$\textcolor{w h i t e}{x + i y} = \frac{5 + 14 i}{4 + 9}$

$\textcolor{w h i t e}{x + i y} = \frac{5 + 14 i}{13}$

$\textcolor{w h i t e}{x + i y} = \frac{5}{13} + \frac{14}{13} i$

So $x = \frac{5}{13} \text{ }$ and $\text{ } y = \frac{14}{13}$