To solve the given system of equations involving complex numbers z and w , we need to find these numbers in the form x + y\imaginaryI , where x, y \in \mathbb{Z} .

The system of equations is:

\begin{cases} w^* - 3z^* = -5 - \imaginaryI \\ \frac{w}{z^*} = -\imaginaryI \end{cases}

Step 1: Express w in terms of z^*

From the second equation:

\frac{w}{z^*} = -\imaginaryI \implies w = -\imaginaryI z^*

Step 2: Substitute w into the first equation

Since w = -\imaginaryI z^* , the conjugate w^* is:

w^* = (-\imaginaryI z^*)^* = ( -\imaginaryI )^* ( z^* )^* = \imaginaryI z

Substitute w^* = \imaginaryI z into the first equation :

\imaginaryI z - 3 z^* = -5 - \imaginaryI

Step 3: Express z in terms of real and imaginary parts

Let z = x + y\imaginaryI , where x, y \in \mathbb{Z} . Then z^* = x - y\imaginaryI .

Substitute z and z^* into the equation:

\imaginaryI(x + y\imaginaryI) - 3(x - y\imaginaryI) = -5 - \imaginaryI

Simplify the left side:

Multiply \imaginaryI(x + y\imaginaryI) :
\imaginaryI x + \imaginaryI^2 y = \imaginaryI x - y
Expand -3(x - y\imaginaryI) :
-3x + 3y\imaginaryI
Combine the results:
(\imaginaryI x - y) + (-3x + 3y\imaginaryI) = (-y - 3x) + \imaginaryI(x + 3y)

The equation becomes:

(-y - 3x) + \imaginaryI(x + 3y) = -5 - \imaginaryI

Step 4: Equate the real and imaginary parts

Set the real parts equal:

-y - 3x = -5 \quad \text{(Equation A)}

Set the imaginary parts equal:

x + 3y = -1 \quad \text{(Equation B)}

Step 5: Solve the system of linear equations

Adding 3 times Equation A and Equation B :

3(-y - 3x) + (x + 3y) = 3(-5) - 1

-3y - 9x + x + 3y = -16

-8x = -16 \implies x = 2

Substitute into Equation A:

-y - 3(2) = -5

-y - 6 = -5

-y = 1 \implies y = -1

Step 6: Find z and w

z = x + y\imaginaryI = 2 - \imaginaryI
z^* = 2 + \imaginaryI

Recall that w = -\imaginaryI z^* :

w = -\imaginaryI(2 + \imaginaryI) = -\imaginaryI \cdot 2 - \imaginaryI \cdot \imaginaryI = -2\imaginaryI + 1

Thus,

w = 1 - 2\imaginaryI

Answer

\boxed{ \begin{cases} z = 2 - \imaginaryI \\ w = 1 - 2\imaginaryI \end{cases} }

Both z and w have integer real and imaginary parts, as required.

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