Question

select the correct answer from the drop-down menu. find the missing term. ( ) + (7 - 3i) + (5 + 9i) + 13i = 10 - 5i. the options in the drop-down are: -2 - 24i, 2 - 12i, -12 - 24i, -2 + 14i

Explanation:

Step1: Let the missing term be \(a + bi\)

We have the equation \((a + bi)+(7 - 3i)+(5 + 9i)+13i = 10 - 5i\)

Step2: Combine like terms (real and imaginary)

First, combine the real parts: \(a + 7+5\)
Then, combine the imaginary parts: \(b - 3 + 9+13\)
So the equation becomes \((a + 12)+(b + 19)i=10 - 5i\)

Step3: Equate real and imaginary parts

For real parts: \(a + 12 = 10\), solving for \(a\) gives \(a=10 - 12=- 2\)
For imaginary parts: \(b + 19=-5\), solving for \(b\) gives \(b=-5 - 19=-24\)
So the missing term is \(-2-24i\)
Wait, there is a mistake above. Let's redo step 2.
Wait, the imaginary parts: \(b-3 + 9+13=b+( - 3+9 + 13)=b + 19\)? No, \(-3i+9i + 13i=(-3 + 9+13)i=19i\)? Wait no, \(-3 + 9=6\), \(6 + 13 = 19\)? Wait, no, the original equation is \((a + bi)+(7 - 3i)+(5 + 9i)+13i=10 - 5i\)
Let's add the known complex numbers: \((7 - 3i)+(5 + 9i)+13i=7 + 5+(-3i + 9i+13i)=12+( - 3 + 9+13)i=12 + 19i\)? Wait, but the right side is \(10 - 5i\), so \((a + bi)+12 + 19i=10 - 5i\)
Then \(a + bi=10 - 5i-(12 + 19i)=10-12+(-5i-19i)=-2-24i\)
Wait, but let's check the options. One of the options is \(-2 - 24i\)
Wait, let's re - calculate the imaginary parts again.
\(-3i+9i + 13i=(-3 + 9+13)i=(6 + 13)i = 19i\). Then the sum of the known complex numbers (excluding the missing term) is \(7 + 5+19i=12 + 19i\)
We know that \((missing\ term)+12 + 19i=10 - 5i\)
So \(missing\ term=10 - 5i-(12 + 19i)=10-12+(-5 - 19)i=-2-24i\)

Answer:

\(-2 - 24i\)