The system of equations you provided is:

$-4x + y = 19$ $3x - 2y = -18$

To solve this using the inverse matrix method, we will first write the system of equations in matrix form $A \cdot X = B$, where:

$A = \begin{pmatrix} -4 & 1 \\ 3 & -2 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \end{pmatrix}, \quad B = \begin{pmatrix} 19 \\ -18 \end{pmatrix}$

To solve for $X$, we use the formula:

$X = A^{-1} \cdot B$

First, we will calculate the inverse of matrix $A$, $A^{-1}$. The formula for the inverse of a 2x2 matrix is:

$A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ where the matrix $A$ is given by:

$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$

For your matrix:

$A = \begin{pmatrix} -4 & 1 \\ 3 & -2 \end{pmatrix}$

The determinant $\text{det}(A)$ is calculated as:

$\text{det}(A) = (a \cdot d) - (b \cdot c)$

Let's calculate the determinant and then find $A^{-1}$. numpy as np

Define the matrix A

A = np.array([[-4, 1], [3, -2]])

Calculate the determinant of A

det_A = np.linalg.det(A)

Calculate the inverse of A

A_inv = np.linalg.inv(A)

det_A, A_invThe determinant of matrix $A$ is approximately 5 (det $A = 5$).

The inverse of matrix $A$ is:

$A^{-1} = \begin{pmatrix} -0.4 & -0.2 \\ -0.6 & -0.8 \end{pmatrix}$

Next, we will multiply $A^{-1}$ by matrix $B$ to solve for the vector $X = \begin{pmatrix} x \\ y \end{pmatrix}$:

$B = \begin{pmatrix} 19 \\ -18 \end{pmatrix}$

Thus, the solution is:

$X = A^{-1} \cdot B$

Let's calculate this multiplication. Define the matrix B B = np.array([19, -18])

Calculate the solution X

X = np.dot(A_inv, B) XThe solution to the system of equations is:

$x = -4, \quad y = 3$

Thus, the values of $x$ and $y$ are $x = -4$ and $y = 3$.

Would you like more details on the solution process, or do you have any other questions? Here are some related questions you might find interesting:

1. How do you verify the solution to a system of linear equations?
2. How do you solve a system of linear equations using Gaussian elimination?
3. What is the geometric interpretation of solving a system of linear equations?
4. What is the significance of the determinant when using matrix inversion?
5. How can you solve larger systems of equations using inverse matrices?

Tip: For larger systems, numerical methods like Gaussian elimination or LU decomposition may be more efficient than using the inverse matrix.