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We are asked to solve the system of equations using the method of matrix inversion. The system is:

\[
\begin{aligned}
5x + 6y &= 39 \\
7x - 8y &= -11
\end{aligned}
\]

This can be written in matrix form as:

\[
\begin{pmatrix} 5 & 6 \\ 7 & -8 \end{pmatrix}
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 39 \\ -11 \end{pmatrix}
\]

Let \( A = \begin{pmatrix} 5 & 6 \\ 7 & -8 \end{pmatrix} \), and the solution vector is \( \mathbf{x} = \begin{pmatrix} x \\ y \end{pmatrix} \), and the constant vector \( \mathbf{b} = \begin{pmatrix} 39 \\ -11 \end{pmatrix} \).

To solve for \( \mathbf{x} \), we use the formula:

\[
\mathbf{x} = A^{-1} \mathbf{b}
\]

### Step 1: Find the inverse of matrix \( A \)

The inverse of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:

\[
A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]

where the determinant \( \text{det}(A) \) is:

\[
\text{det}(A) = ad - bc
\]

For our matrix \( A = \begin{pmatrix} 5 & 6 \\ 7 & -8 \end{pmatrix} \), the determinant is:

\[
\text{det}(A) = (5)(-8) - (6)(7) = -40 - 42 = -82
\]

Thus, the inverse of \( A \) is:

\[
A^{-1} = \frac{1}{-82} \begin{pmatrix} -8 & -6 \\ -7 & 5 \end{pmatrix}
\]

\[
A^{-1} = \begin{pmatrix} \frac{8}{82} & \frac{6}{82} \\ \frac{7}{82} & \frac{-5}{82} \end{pmatrix}
\]
\[
A^{-1} = \begin{pmatrix} \frac{4}{41} & \frac{3}{41} \\ \frac{7}{82} & \frac{-5}{82} \end{pmatrix}
\]

### Step 2: Multiply \( A^{-1} \) by \( \mathbf{b} \)

Now, multiply the inverse matrix \( A^{-1} \) by the constant vector \( \mathbf{b} \):

\[
\mathbf{x} = \begin{pmatrix} \frac{4}{41} & \frac{3}{41} \\ \frac{7}{82} & \frac{-5}{82} \end{pmatrix} \begin{pmatrix} 39 \\ -11 \end{pmatrix}
\]

Performing the matrix multiplication:

\[
x = \frac{4}{41} \times 39 + \frac{3}{41} \times (-11) = \frac{156}{41} + \frac{-33}{41} = \frac{123}{41} = 3
\]

\[
y = \frac{7}{82} \times 39 + \frac{-5}{82} \times (-11) = \frac{273}{82} + \frac{55}{82} = \frac{328}{82} = 4
\]

### Final Solution

The solution is:

\[
x = 3, \quad y = 4
\]

Would you like more details or have any questions?

---

Here are some related questions:

1. What is the determinant of a matrix, and how is it used in finding the inverse?
2. How can you verify the solution of a system of equations using matrix multiplication?
3. Can matrix inversion be applied to larger systems of equations?
4. What are the conditions for a matrix to be invertible?
5. What are some alternative methods for solving systems of equations (e.g., Cramer's rule)?

**Tip:** When dealing with matrices, always check if the determinant is zero before attempting to find the inverse—if it is, the matrix is non-invertible.

Solve by the method of matrix inversion:
5x + 6y = 39
7x - 8y = -11

Learn how to solve the system of linear equations 5x + 6y = 39 and 7x - 8y = -11 using the method of matrix inversion. This problem involves key concepts of linear algebra, including matrix inversion and determinants. Suitable for students in Grades 10-12.

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