To solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the equation [tex]\((x + yi) + (4 + 9i) = 9 - 4i\)[/tex], we need to equate the real and imaginary parts on both sides of the equation separately.
The equation given is:
[tex]\[
(x + yi) + (4 + 9i) = 9 - 4i
\][/tex]
First, let's separate the real and imaginary parts.
Real part:
The real part on the left side is [tex]\( x + 4 \)[/tex].
The real part on the right side is [tex]\( 9 \)[/tex].
So, we can set up the equation for the real part:
[tex]\[
x + 4 = 9
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = 9 - 4
\][/tex]
[tex]\[
x = 5
\][/tex]
Imaginary part:
The imaginary part on the left side is [tex]\( yi + 9i \)[/tex].
The imaginary part on the right side is [tex]\( -4i \)[/tex].
So, we can set up the equation for the imaginary part:
[tex]\[
yi + 9i = -4i
\][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[
y + 9 = -4
\][/tex]
[tex]\[
y = -4 - 9
\][/tex]
[tex]\[
y = -13
\][/tex]
Hence, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy the given equation are:
[tex]\[
x = 5 \quad \text{and} \quad y = -13
\][/tex]
Thus, the correct answer is:
[tex]\[
\text{B. } x=5 \text{ and } y=-13
\][/tex]
