https://r2.erweima.ai/mathbot/image/1726687172572etylepa3.jpg

Resuelve las siguientes sistemas de ecuaciones por medio de la inversa como te lo plantea el problema 

I have displayed the image with the system of equations. Let's solve these systems using the inverse matrix method.

### First System:
\[
\begin{aligned}
x - 2y + 4z &= 12 \\
2x - y + 5z &= 18 \\
-x + 3y - 3z &= -8
\end{aligned}
\]

This can be written in matrix form as:

\[
AX = B
\]

where:

\[
A = \begin{pmatrix} 1 & -2 & 4 \\ 2 & -1 & 5 \\ -1 & 3 & -3 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad B = \begin{pmatrix} 12 \\ 18 \\ -8 \end{pmatrix}
\]

We need to find \(X = A^{-1} B\), which is the solution for \(x\), \(y\), and \(z\).

### Second System:
\[
\begin{aligned}
x - y + 3z &= 8 \\
2x - y + 4z &= 11 \\
-x + 2y - 4z &= -11
\end{aligned}
\]

This is also written in matrix form:

\[
A = \begin{pmatrix} 1 & -1 & 3 \\ 2 & -1 & 4 \\ -1 & 2 & -4 \end{pmatrix}, \quad B = \begin{pmatrix} 8 \\ 11 \\ -11 \end{pmatrix}
\]

### Third System:
\[
\begin{aligned}
x + 2y - 3z &= 11 \\
3x + 6y - 8z &= 32 \\
-2x - y &= -7
\end{aligned}
\]

This matrix form is:

\[
A = \begin{pmatrix} 1 & 2 & -3 \\ 3 & 6 & -8 \\ -2 & -1 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 11 \\ 32 \\ -7 \end{pmatrix}
\]

Let's compute the solutions for all three systems.Here are the solutions for the systems of equations using the inverse matrix method:

1. **First system:**
   \[
   x = 2, \quad y = 1, \quad z = 3
   \]

2. **Second system:**
   \[
   x = 1, \quad y = -1, \quad z = 2
   \]

3. **Third system:**
   \[
   x = 2, \quad y = 3, \quad z = -1
   \]

Would you like any further details or have any questions?

### Related Questions:
1. How do you calculate the inverse of a matrix?
2. Why is the inverse method used to solve systems of equations?
3. What happens if the determinant of the matrix is zero?
4. How does Gaussian elimination compare to the inverse method?
5. Can we solve larger systems (e.g., 4x4 or higher) using this method?

### Tip:
When using the inverse matrix method, ensure that the matrix is invertible by checking that its determinant is non-zero before proceeding with the solution.

Encuentra la solución de los siguientes sistemas de ecuaciones por medio de la inversa:

Learn how to solve systems of equations using the inverse matrix method. This problem involves three linear systems, where the matrices are invertible. Suitable for students in high school or undergraduate math courses.

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