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To solve the problem, let's interpret the given matrices and perform the necessary operations.

### Given:
- \( A = \begin{bmatrix} 1 & 2 \\ 3 & -1 \end{bmatrix} \)
- \( B = \begin{bmatrix} x \\ y \end{bmatrix} \)
- \( AB = \begin{bmatrix} 6 \\ 8 \end{bmatrix} \)

### Problem:
We need to determine the values of \(x\) and \(y\).

### Step-by-Step Solution:

1. **Matrix Multiplication:**

   To find the product \( AB \), we multiply matrix \( A \) by matrix \( B \):

   \[
   AB = \begin{bmatrix} 1 & 2 \\ 3 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}
   \]

2. **Calculate Each Element of the Resulting Matrix:**

   The multiplication results in a 2x1 matrix:

   \[
   AB = \begin{bmatrix} 1 \cdot x + 2 \cdot y \\ 3 \cdot x + (-1) \cdot y \end{bmatrix}
   = \begin{bmatrix} x + 2y \\ 3x - y \end{bmatrix}
   \]

3. **Set Up Equations:**

   We know \( AB = \begin{bmatrix} 6 \\ 8 \end{bmatrix} \), so we equate each element:

   \[
   \begin{cases}
   x + 2y = 6 \\
   3x - y = 8
   \end{cases}
   \]

4. **Solve the System of Equations:**

   Let's solve the first equation for \(x\):

   \[
   x = 6 - 2y
   \]

   Substitute this into the second equation:

   \[
   3(6 - 2y) - y = 8
   \]

   Simplify:

   \[
   18 - 6y - y = 8
   \]

   \[
   18 - 7y = 8
   \]

   \[
   -7y = 8 - 18
   \]

   \[
   -7y = -10
   \]

   \[
   y = \frac{-10}{-7} = \frac{10}{7}
   \]

5. **Find \(x\):**

   Substitute \( y = \frac{10}{7} \) into \( x = 6 - 2y \):

   \[
   x = 6 - 2 \left(\frac{10}{7}\right)
   \]

   \[
   x = 6 - \frac{20}{7}
   \]

   \[
   x = \frac{42}{7} - \frac{20}{7}
   \]

   \[
   x = \frac{22}{7}
   \]

### Conclusion:

The values of \( x \) and \( y \) are:

\[
x = \frac{22}{7}, \quad y = \frac{10}{7}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you multiply a matrix by a vector?
2. What is the determinant of matrix \(A\)?
3. How can we verify the solution for \(x\) and \(y\)?
4. What are some applications of solving systems of linear equations using matrices?
5. What is the significance of the inverse of a matrix in solving systems of equations?

### Tip:
Always double-check your matrix multiplication steps to ensure accuracy in solving for unknowns.

Learn how to solve a system of linear equations involving matrices and vectors with detailed step-by-step solutions. This problem involves matrix multiplication and solving simultaneous equations. Suitable for students in Grades 10-12.

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