Question

M e N são reais que satisfazem a igualdade 5i – 3(M – Ni) + 2i(M + Ni) = 0. Calcule M + N

Answer 1

$5i-3\cdot(\text{M}-\text{N}\cdot i)+2i\cdot(\text{M}+\text{N}\cdot i)=0$

$5i-3\text{M}+3\text{N}\cdot i+2\text{M}\cdot i+2\text{N}\cdot i^2=0$

$5i-3\text{M}+3\text{N}\cdot i+2\text{M}\cdot i-2\text{N}=0$

$-3\text{M}-2\text{N}+(5+3\text{N}+2\text{M})\cdot i=0$

Logo, devemos ter:

$\begin{cases}-3\text{M}-2\text{N}=0\\5+3\text{N}+2\text{M}=0 \end{cases} \iff \begin{cases}3\text{M}+2\text{N}=0 \\ 2\text{M}+3\text{N}=-5 \end{cases}$

Multiplicando a primeira equação por $-2$ e a segunda por $3$:

$\begin{cases}3\text{M}+2\text{N}=0 ~~~~~~~\times(-2) \\ 2\text{M}+3\text{N}=-5~~~~\times3 \end{cases} \iff \begin{cases}-6\text{M}-4\text{N}=0 \\ 6\text{M}+9\text{N}=-15 \end{cases}$

Somando membro a membro:

$-6\text{M}-4\text{N}+6\text{M}+9\text{N}=-15$

$5\text{N}=-15$

$\text{N}=\dfrac{-15}{5}$

$\boxed{\text{N}=-3}$

Substituindo na primeira equação:

$3\text{M}+2\text{N}=0$

$3\text{M}+2\cdot(-3)=0$

$3\text{M}-6=0$

$3\text{M}=6$

$\text{M}=\dfrac{6}{3}$

$\boxed{\text{M}=2}$

Logo, $\text{M}+\text{N}=-3+2 \iff \boxed{\text{M}+\text{N}=-1}$