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Para resolver el sistema de ecuaciones utilizando la regla de Cramer, primero necesitamos encontrar el determinante de la matriz de coeficientes (denotado como \( D \)) y los determinantes de las matrices modificadas para cada incógnita (denotadas como \( D_{x_1}, D_{x_2}, D_{x_3} \)).

El sistema es:

\[
5x_1 - x_2 - 4x_3 = 30
\]
\[
-2x_1 + 3x_2 + x_3 = -5
\]
\[
-4x_1 + 2x_2 + 6x_3 = -30
\]

1. **Matriz de coeficientes:**

\[
A = \begin{pmatrix}
5 & -1 & -4 \\
-2 & 3 & 1 \\
-4 & 2 & 6
\end{pmatrix}
\]

2. **Determinante de la matriz \( D \):**

Calcularé el determinante de \( A \), luego los determinantes de las matrices modificadas reemplazando las columnas por el vector de términos independientes, y finalmente los valores de \( x_1 \), \( x_2 \) y \( x_3 \). Vamos a ello.It seems like I can’t do more advanced data analysis right now. Please try again later. If you need any clarification about the process of solving the system using the rule of Cramer, feel free to ask!

Here are some related questions you might find helpful:
1. How do you compute the determinant of a 3x3 matrix manually?
2. What is the rule of Cramer and when is it used?
3. Can Cramer's rule be applied to non-square matrices?
4. How would the process change if one of the determinants is zero?
5. What are some alternative methods to solve systems of linear equations?

**Tip**: Cramer's rule is best suited for systems where the matrix of coefficients has a non-zero determinant, ensuring a unique solution exists.

Resuelve las siguientes ecuaciones utilizando la regla de Cramer: 
5x1 - x2 - 4x3 = 30, 
-2x1 + 3x2 + x3 = -5, 
-4x1 + 2x2 + 6x3 = -30.

Learn how to solve the system of linear equations: 5x1 - x2 - 4x3 = 30, -2x1 + 3x2 + x3 = -5, -4x1 + 2x2 + 6x3 = -30 using Cramer's Rule. This step-by-step guide covers key concepts in linear algebra, including determinants and matrix manipulations. Suitable for students in Grades 10-12.

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