Question

let ( z = 1 + 3i ) and ( w - z = - 3 - 4i ). what is ( w )?
( -2 + i )
( -2 - i )
( 4 - 7i )
( 4 + 7i )

Explanation:

Step1: Isolate \( w \)

We have the equation \( w - z = -3 - 4i \). To find \( w \), we add \( z \) to both sides of the equation. So, \( w = z + (-3 - 4i) \).

Step2: Substitute \( z = 1 + 3i \)

Substitute \( z = 1 + 3i \) into the equation for \( w \). We get \( w=(1 + 3i)+(-3 - 4i) \).

Step3: Combine like terms

For the real parts: \( 1+(-3)=1 - 3=-2 \). For the imaginary parts: \( 3i+(-4i)=3i - 4i=-i \). So, \( w=-2 - i \)? Wait, no, wait, let's check again. Wait, \( (1 + 3i)+(-3 - 4i)=(1-3)+(3i - 4i)=-2 - i \)? But that's not matching. Wait, maybe I made a mistake. Wait, the equation is \( w - z=-3 - 4i \), so \( w=z+(-3 - 4i) \), \( z = 1+3i \), so \( w=(1 + 3i)+(-3 - 4i)=1-3+3i - 4i=-2 - i \)? But the options have -2 -i as an option (second option). Wait, let's verify:

If \( w=-2 - i \), then \( w - z=(-2 - i)-(1 + 3i)=-2 - i -1 - 3i=-3 - 4i \), which matches the given equation. So the correct value of \( w \) is \( -2 - i \).

Answer:

B. \( -2 - i \)