Question

Determine os números complexos z1=1+zi, z2=-1+3i e z3=z-zi,Calculem:
3z+4i=2-6i

Answer 1

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Determinar os números complexos

•   $\mathsf{z_1=1+zi}$

•   $\mathsf{z_1=-1+3i}$

•   $\mathsf{z_3=z-zi}$


dado que $\mathsf{z}$ satisfaz a seguinte igualdade:

$\mathsf{3z+4i=2-6i}\\\\ \mathsf{3z=2-6i-4i}\\\\ \mathsf{3z=2-10i}\\\\ \mathsf{z=\dfrac{2-10i}{3}}\\\\\\ \mathsf{z=\dfrac{2}{3}-\dfrac{10}{3}\,i\qquad\quad\checkmark}$


Dessa forma temos que

•   $\mathsf{z_1=1+zi}$

$\mathsf{z_1=1+\bigg(\dfrac{2}{3}-\dfrac{10}{3}\,i\bigg)\cdot i}\\\\\\ \mathsf{z_1=1+\dfrac{2}{3}\,i-\dfrac{10}{3}\,i^2}\\\\\\ \mathsf{z_1=1+\dfrac{2}{3}\,i-\dfrac{10}{3}\cdot (-1)}\\\\\\ \mathsf{z_1=1+\dfrac{2}{3}\,i+\dfrac{10}{3}}$

$\mathsf{z_1=\dfrac{3}{3}+\dfrac{2}{3}\,i+\dfrac{10}{3}}\\\\\\ \mathsf{z_1=\dfrac{3+10}{3}+\dfrac{2}{3}\,i}\\\\\\ \mathsf{z_1=\dfrac{13}{3}+\dfrac{2}{3}\,i\qquad\quad\checkmark}$


•   $\mathsf{z_2=-1+3i}$

Este número já está determinado, ele não depende de $\mathsf{z}.$


•   $\mathsf{z_3=z-zi}$

$\mathsf{z_3=\bigg(\dfrac{2}{3}-\dfrac{10}{3}\,i\bigg)-\bigg(\dfrac{2}{3}-\dfrac{10}{3}\,i\bigg)\cdot i}\\\\\\ \mathsf{z_3=\dfrac{2}{3}-\dfrac{10}{3}\,i-\dfrac{2}{3}\,i+\dfrac{10}{3}\,i^2}\\\\\\ \mathsf{z_3=\dfrac{2}{3}-\dfrac{10+2}{3}\,i+\dfrac{10}{3}\cdot (-1)}\\\\\\ \mathsf{z_3=\dfrac{2}{3}-\dfrac{12}{3}\,i-\dfrac{10}{3}}$

$\mathsf{z_3=\dfrac{2-10}{3}-\dfrac{12}{3}\,i}\\\\\\ \mathsf{z_3=-\,\dfrac{8}{3}-4i\qquad\quad\checkmark}$


Bons estudos! :-)