The uploaded image contains a system of linear equations involving complex numbers, and the task is to determine which of the given options is the correct solution for the system. Let me extract the key mathematical content and work on it.

The equations provided are:

1. $(-2i) \cdot x + (-2 + 6i) \cdot y + (-2 - 2i) \cdot z = -14 - 14i$
2. $(2 + 3i) \cdot x + (-3 - 11i) \cdot y + (-2 - 3i) \cdot z = 7 - 9i$

The options for solutions are:

3 + i \ 1 \ 0 \end{bmatrix}$2.$\begin{bmatrix} 3 - 3i \ 0 \ 4 \end{bmatrix}$3.$\begin{bmatrix} 8 + 2i \ 2 + i \ 4 \end{bmatrix}$$

Let me calculate and verify which option satisfies the system of equations. sympy as sp

Define variables

x, y, z = sp.symbols('x y z')

Define the system of equations (using complex numbers)

eq1 = sp.Eq((-2j)*x + (-2 + 6j)*y + (-2 - 2j)*z, -14 - 14j) eq2 = sp.Eq((2 + 3j)*x + (-3 - 11j)*y + (-2 - 3j)*z, 7 - 9j)

Define the solution options

option1 = {x: 3 + 1j, y: 1, z: 0} option2 = {x: 3 - 3j, y: 0, z: 4} option3 = {x: 8 + 2j, y: 2 + 1j, z: 4}

Verify each option

result1_eq1 = eq1.subs(option1) result1_eq2 = eq2.subs(option1)

result2_eq1 = eq1.subs(option2) result2_eq2 = eq2.subs(option2)

result3_eq1 = eq1.subs(option3) result3_eq2 = eq2.subs(option3)

result1_eq1, result1_eq2, result2_eq1, result2_eq2, result3_eq1, result3_eq2After substituting each option into the equations, the results indicate that further simplifications are necessary to confirm which solution works. Let me simplify these equations step-by-step to determine the correct answer. Simplify results for each option result1_eq1_simplified = sp.simplify(result1_eq1.lhs - result1_eq1.rhs) result1_eq2_simplified = sp.simplify(result1_eq2.lhs - result1_eq2.rhs)

result2_eq1_simplified = sp.simplify(result2_eq1.lhs - result2_eq1.rhs) result2_eq2_simplified = sp.simplify(result2_eq2.lhs - result2_eq2.rhs)

result3_eq1_simplified = sp.simplify(result3_eq1.lhs - result3_eq1.rhs) result3_eq2_simplified = sp.simplify(result3_eq2.lhs - result3_eq2.rhs)

result1_eq1_simplified, result1_eq2_simplified, result2_eq1_simplified, result2_eq2_simplified, result3_eq1_simplified, result3_eq2_simplifiedAfter simplification, the results are as follows:

1. Option 1:

   • Equation 1: Residual = $14 + 14i$ (not satisfied).
   • Equation 2: Residual = $-7 + 9i$ (not satisfied).

2. Option 2:

   • Equation 1: Residual = $0$ (satisfied).
   • Equation 2: Residual = $0$ (satisfied).

3. Option 3:

   • Equation 1: Residual = $0$ (satisfied).
   • Equation 2: Residual = $0$ (satisfied).

Both Option 2 and Option 3 satisfy the system of equations. This might require further clarification from the question to determine additional conditions (e.g., preference for specific solutions).

Would you like me to explore this further or confirm other details?

Here are 5 follow-up questions based on this:

1. Would you like to verify the method of substitution for each solution step-by-step?
2. Should I clarify why Option 1 does not satisfy the system of equations?
3. Would you like a detailed explanation of how complex arithmetic works in solving these systems?
4. Should we consider any constraints to choose between Option 2 and Option 3?
5. Would you like help solving a similar system of equations?

Tip: When solving systems with complex numbers, always verify both the real and imaginary parts separately to ensure correctness.