To find the sum of the complex numbers [tex]\(12 - 5i\)[/tex] and [tex]\(-3 + 4i\)[/tex], we will add their real parts and their imaginary parts separately.
1. Identify the complex numbers:
[tex]\[
z_1 = 12 - 5i
\][/tex]
[tex]\[
z_2 = -3 + 4i
\][/tex]
2. Add the real parts:
The real part of [tex]\(z_1\)[/tex] is 12.
The real part of [tex]\(z_2\)[/tex] is -3.
[tex]\[
\text{Real part: } 12 + (-3) = 9
\][/tex]
3. Add the imaginary parts:
The imaginary part of [tex]\(z_1\)[/tex] is -5i.
The imaginary part of [tex]\(z_2\)[/tex] is 4i.
[tex]\[
\text{Imaginary part: } -5i + 4i = -1i \text{ or } -i
\][/tex]
4. Combine the results:
[tex]\[
(12 - 5i) + (-3 + 4i) = 9 - i
\][/tex]
So, the sum of [tex]\(12 - 5i\)[/tex] and [tex]\(-3 + 4i\)[/tex] is [tex]\(9 - i\)[/tex].
Now, let's compare this result with the given options.
- [tex]\(-16 + 63i\)[/tex]
- [tex]\(9 - i\)[/tex]
- [tex]\(9 - 9i\)[/tex]
- [tex]\(15 - 9i\)[/tex]
The correct answer is [tex]\(9 - i\)[/tex].
