The equation in the image is:

$\overline{z} + i|z| = 12 + 9i$

Where:

• $z$ is a complex number,
• $\overline{z}$ is the complex conjugate of $z$,
• $i$ is the imaginary unit,
• $|z|$ is the modulus (magnitude) of $z$.

Let $z = x + yi$, where $x$ is the real part and $y$ is the imaginary part of the complex number. Therefore:

• $\overline{z} = x - yi$ (complex conjugate of $z$),
• $|z| = \sqrt{x^2 + y^2}$ (modulus of $z$).

Now, let's substitute these into the equation:

$(x - yi) + i\sqrt{x^2 + y^2} = 12 + 9i$

Separate the real and imaginary parts:

• Real part: $x = 12$,
• Imaginary part: $-y + \sqrt{x^2 + y^2} = 9$.

Since $x = 12$, substitute it into the imaginary part equation:

$-y + \sqrt{12^2 + y^2} = 9$

$-y + \sqrt{144 + y^2} = 9$

Now, let's solve for $y$.The solution for $y$, the imaginary part of $z$, is:

$y = \frac{7}{2} = 3.5$

So, the value of the imaginary part of $z$ is $3.5$.

Would you like any further details or explanations?

Here are 5 related questions to expand on this concept:

1. How do you calculate the modulus of a complex number?
2. What is the significance of the complex conjugate in equations like this?
3. How can you apply this method to solve more general complex number equations?
4. Why is separating real and imaginary parts essential when solving complex equations?
5. What are some real-world applications of complex numbers?

Tip: Always ensure you correctly separate real and imaginary parts when solving equations involving complex numbers.