Question

if z=a+bi and -z=a-bi,
solve z+6(-z)=7 for z

Answer 1

hint:
if
z= a+ bi, then:
-z = -a - bi

Answer 2

furthermore:
z-6z=-5z

Answer 3

sorry its actually the conjugate of z, not -z so its a-bi

Answer 4

ok!
then your equation can be rewritten as follows:
\[\Large z + 6\bar z = a + bi + 6a - 6bi = 7a - 5bi = 7\]

Answer 5

which is equivalent to these two real equations:
\[\Large \left\{ \begin{gathered}
7a = 7 \hfill \\
- 5b = 0 \hfill \\
\end{gathered} \right.\]
please solve that system for a and b

Answer 6

a=1 and b=0 ?

Answer 7

that's right!
so, what is z=...?

Answer 8

do i just plug these numbers into 7a-5bi?

Answer 9

no, you have to plug, those values, for a and b, into z=a+bi

Answer 10

what do you get?

Answer 11

so its 1?

Answer 12

correct!
z=1 is the solution of your complex equation

Answer 13

OH ok! Thanks so much!

Answer 14

:)

Answer 15

question, where does the i go?
my teacher asked and i wasn't sure