Question

Determine x e y reais de modo que: (x + yi)* (3 - 2i)= 4 + 5i.

Answer 1

$(x + yi)* (3 - 2i)= 4 + 5i \\ \\ \\ (x + yi)= \dfrac{4+5i}{3-2i} \\ \\ \\ (x + yi)= \dfrac{(4+5i)}{(3-2i)} * \dfrac{(3+2i)}{(3+2i) } \\ \\ \\ (x + yi)= \dfrac{4*3+4*2i+5i*3+5*2i^2}{3^2-(2i)^2}\qquad i^2=-1 \\ \\ \\ (x + yi)= \dfrac{12+8i+15i+10*(-1)}{9-(4*(-1))} \\ \\ \\ (x + yi)= \dfrac{12+8i+15i-10}{9-(-4)} \\ \\ \\ (x + yi)= \dfrac{2+23i}{9+4} \\ \\ \\ (x + yi)= \dfrac{2+23i}{13} \\ \\ \\ \boxed{ (x + yi)= \dfrac{2}{13}+ \dfrac{23}{13}i}$

$\boxed{ x= \frac{2}{13}\qquad y= \frac{23}{13} } }$

Answer 2

 (x + yi)* (3 - 2i)= 4 + 5i.

3x - 2xi + 3yi -  2yi^2 =  4 + 5i ==> 3x - 2xi + 3yi -  2y(-1) = 4 + 5i

3x - 2xi + 3yi +  2y = 4 + 5i ==>  3x + 2y + ( - 2x + 3y )i  = 4 + 5i

  3x + 2y = 4(2)   
- 2x + 3y = 5(3)


   6x + 4y = 8
- 6x + 9y = 15
        13y = 23    ==> y = 23/13

3x + 2y = 4 ==> 3x = 4 - 2y ==> 3x = 4 - 2(23/13)

3x = 4 - 46  ==> 39x = 52 - 46 ==> 39x = 6  ==> x = 6/39  ou x = 2/13 
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