To determine the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy the equation [tex]\((x + yi) + (4 + 9i) = 9 - 4i\)[/tex], we can separate the equation into real and imaginary parts and solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
1. Start with the given equation:
[tex]\[
(x + yi) + (4 + 9i) = 9 - 4i
\][/tex]
2. Combine the real parts and the imaginary parts separately:
[tex]\[
(x + 4) + (y + 9)i = 9 - 4i
\][/tex]
3. Set the real parts equal to each other and the imaginary parts equal to each other:
[tex]\[
x + 4 = 9 \quad \text{and} \quad y + 9 = -4
\][/tex]
4. Solve for [tex]\( x \)[/tex] from the real parts equation:
[tex]\[
x + 4 = 9
\][/tex]
[tex]\[
x = 9 - 4
\][/tex]
[tex]\[
x = 5
\][/tex]
5. Solve for [tex]\( y \)[/tex] from the imaginary parts equation:
[tex]\[
y + 9 = -4
\][/tex]
[tex]\[
y = -4 - 9
\][/tex]
[tex]\[
y = -13
\][/tex]
Therefore, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy the equation are [tex]\( x = 5 \)[/tex] and [tex]\( y = -13 \)[/tex].
The correct answer is:
C. [tex]\(x=5\)[/tex] and [tex]\(y=-13\)[/tex]
