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To express this system of equations as a matrix equation \( AX = B \), let’s start by identifying the coefficients and constants.

The system given is:
\[
\begin{cases} 
x + 5y = 7 \\ 
-6x + 8y = 34 
\end{cases}
\]

### Step 1: Identify the Coefficient Matrix \( A \)
The coefficient matrix \( A \) consists of the coefficients of \( x \) and \( y \) from each equation:
\[
A = \begin{bmatrix} 1 & 5 \\ -6 & 8 \end{bmatrix}
\]

### Step 2: Define the Variable Matrix \( X \)
The variable matrix \( X \) contains the variables \( x \) and \( y \):
\[
X = \begin{bmatrix} x \\ y \end{bmatrix}
\]

### Step 3: Define the Constant Matrix \( B \)
The constant matrix \( B \) is made up of the constants from the right side of each equation:
\[
B = \begin{bmatrix} 7 \\ 34 \end{bmatrix}
\]

### Matrix Equation Representation
Now we can write the system as a matrix equation \( AX = B \):
\[
\begin{bmatrix} 1 & 5 \\ -6 & 8 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 7 \\ 34 \end{bmatrix}
\]

This matrix equation represents the original system of equations.

Would you like further details on how to solve this matrix equation?

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### Related Questions to Explore
1. How can we solve the matrix equation \( AX = B \) using matrix inversion?
2. What methods are available to determine if a system of linear equations is consistent?
3. How does Gaussian elimination work to solve a system of linear equations?
4. How would you interpret the determinant of the coefficient matrix \( A \) in this context?
5. Can a system of equations have infinitely many solutions, and how would that look in matrix form?

### Tip:
When converting a system of equations into a matrix form, always align the variables in the same order in each equation to ensure correct matrix representation.

Write the system of equations as a matrix equation AX = B, with A as the coefficient matrix of the system: x + 5y = 7, -6x + 8y = 34.

Learn how to represent a system of linear equations as a matrix equation in the form AX = B. This example uses the system x + 5y = 7 and -6x + 8y = 34 to illustrate matrix representation. Suitable for high school students in Grades 10-12.

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