Find the missing complex number.
-$$( 7 - 9 \mathrm { i } ) ( ? ) = 9 + 2 \mathrm { i }$$
A) $\frac { 81 } { 26 } + \frac { 67 } { 26 } \mathrm { i }$
B) $- \frac { 9 } { 32 } + \frac { 19 } { 32 } \mathrm { i }$
C) $- \frac { 81 } { 32 } + \frac { 19 } { 32 } \mathrm { i }$
D) $\frac { 9 } { 26 } + \frac { 19 } { 26 } \mathrm { i }$

Question

Quizplus · Accepted Answer

To find the missing complex number, we can set up the equation:
$$(7-9i)(a+bi) = 9+2i$$

Expanding the left-hand side using FOIL method, we get:
$$7a + 7bi - 9ai - 9bi^2 = 9+2i$$

Since $i^2=-1$, we can simplify the equation as follows:
$$(7a+9b) + (7b-9a)i = 9 + 2i$$
Equating the real and imaginary parts on both sides, we get the following system of equations:
\begin{align*}
7a + 9b &= 9 \
7b - 9a &= 2
\end{align*}
Solving for $a$ and $b$, we get:
$$a = \frac{9}{26}, \quad b = \frac{19}{26}$$
Therefore, the missing complex number is:
$$\frac{9}{26} + \frac{19}{26}i$$
which is choice D.

Find the Missing Complex Number (7−9i)(?)=9+2i( 7 - 9 \mathrm { i } ) ( ? ) = 9 + 2 \mathrm { i }(7−9i)(?)=9+2i

Find the Missing Complex Number $( 7 - 9 \mathrm { i } ) ( ? ) = 9 + 2 \mathrm { i }$