Question

if (x + iy)( 4 - 5i) = ( 3 + i) then what will be tne values of x and y

Answer 1

Given ,

(x + iy)( 4 - 5i) = ( 3 + i)

$\implies \: \bold{x + iy = \frac{3 + i}{4 - 5i} }$

$\implies \: \bold{x + iy = \frac{(3 + i)(4 + 5i)}{(4 - 5i)(4 + 5i)} }$

$\implies \: \bold{x + iy = \frac{3 (4 + 5i) + i(4 + 5i)}{ {4}^{2} - {(5i)}^{2} } }$

$\implies \: \bold{x + iy = \frac{12 + 15i + 4i + 5 {i}^{2} }{ 16- {25i}^{2} } }$

$\implies \: \bold{x + iy = \frac{12 + 19i+ 5 ( - 1) }{ 16- {25}( - 1) } }$

$\implies \: \bold{x + iy = \frac{12 + 19i - 5 }{ 16 + 25 } }$

$\implies \: \bold{x + iy = \frac{7+ 19i }{ 41 } }$

$\implies \: \bold{x + iy = \frac{7}{41} + i \frac{19}{41} }$

Comparing the real and imaginary part , we have

$★ \: \underbrace{ \color{orange}{ \bold{ \: \: x = \frac{7}{41} \: \: \: \: and \: \: \: \: y = \frac{19}{41} }}} \: \: ★$