The problem shown is a system of linear equations represented in matrix form. It appears as follows:

$\begin{pmatrix} 4 & 3 \\ 1 & -2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 13 \\ 6 \end{pmatrix}$

We need to solve for the vector $\begin{pmatrix} x \\ y \end{pmatrix}$, which I'll denote as matrix $A$.

Step-by-Step Solution:

1. Expand the matrix equation into individual equations: $4x + 3y = 13$ $x - 2y = 6$

2. Solve the second equation for $x$: $x = 6 + 2y$

3. Substitute $x = 6 + 2y$ into the first equation: $4(6 + 2y) + 3y = 13$ $24 + 8y + 3y = 13$ $24 + 11y = 13$ $11y = -11$ $y = -1$

4. Substitute $y = -1$ back into the expression for $x$: $x = 6 + 2(-1)$ $x = 6 - 2 = 4$

Solution:

$A = \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 4 \\ -1 \end{pmatrix}$

Therefore, the values of $x$ and $y$ that satisfy the system are $x = 4$ and $y = -1$.

Would you like further details on any step or have any questions?

Related Questions:

1. How do you solve a system of linear equations using matrix inversion?
2. What are other methods to solve systems of equations, such as Gaussian elimination?
3. How does Cramer's Rule apply to solving 2x2 linear systems?
4. How can you determine if a system of linear equations has a unique solution, infinitely many solutions, or no solution?
5. What are some practical applications of solving systems of linear equations in real-world problems?

Tip: When working with matrices, always check if the determinant is non-zero to ensure a unique solution exists for systems of equations.