Question

1- Sejam os complexos
$z1 = (2x + 1) + yi \: e \: z2 = - y + 2i$
Determine x e y de modo que Z1 + Z2 = 0

Answer 1

$Z_1+Z_2~=~(2x+1)+yi+(-y)+2i\\\\\\Z_1+Z_2~=~(2x+1)+yi-y+2i\\\\\\\boxed{Z_1+Z_2~=~(2x+1-y)+i\,.\,(y+2)}\\\\\\\boxed{Re(Z_1+Z_2)~=~2x+1-y}\\\\\\\boxed{Im(Z_1+Z_2)~=~y+2}\\\\\\Como~o~enunciado~diz~que~a~soma~deve~valer~0~(complexo),~ou~seja,~\\tanto~a~parte~real~quanto~a~parte~imaginaria~devem~valer~0,~logo:\\\\\\\left\{\begin{array}{ccc}2x+1-y&=&0\\y+2&=&0\end{array}\right\\\\\\Pela~2^a~equacao,~temos:\\\\\\\boxed{y~=~-2}\\\\\\Substituindo~o~valor~de~"y"~na~1^a~equacao:$

$2x+1-(-2)~=~0\\\\\\2x+1+2~=~0\\\\\\2x~=~-3\\\\\\\boxed{x~=~-\dfrac{3}{2}~~ou~\,-1,5}$