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The problem involves solving a system of linear differential equations with initial conditions:

\[
\frac{dx}{dt} = 3x + 8y \quad \text{and} \quad \frac{dy}{dt} = -x - 3y
\]

with the initial conditions \( x(0) = 6 \) and \( y(0) = -2 \).

### Solution Outline

This is a system of first-order linear differential equations, which can be solved using matrix methods or decoupling through eigenvalues and eigenvectors.

1. **Write the system in matrix form**:
   \[
   \frac{d}{dt} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 3 & 8 \\ -1 & -3 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}
   \]

2. **Find the eigenvalues and eigenvectors** of the coefficient matrix:
   \[
   A = \begin{pmatrix} 3 & 8 \\ -1 & -3 \end{pmatrix}
   \]

3. **Solve the characteristic equation**:
   \[
   \text{det}(A - \lambda I) = 0
   \]

4. **Use the eigenvalues and eigenvectors** to express the general solution.

5. **Apply initial conditions** \( x(0) = 6 \) and \( y(0) = -2 \) to find the specific solution.

Let me compute this step by step.The eigenvalues of the matrix \( A \) are \( \lambda_

Determine the solution of the simultaneous equations: dx/dt = 3x + 8y and dy/dt = -x - 3y with initial conditions x(0) = 6, y(0) = -2.

Solve a system of linear differential equations dx/dt = 3x + 8y and dy/dt = -x - 3y with initial conditions x(0) = 6, y(0) = -2. Learn the process of finding eigenvalues and eigenvectors to solve the system.

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