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To solve these sets of equations using the matrix method as requested in the exercise, I will detail each step and calculation required.

### System 1
Given equations:
1. \(x + 3y + 2z = 6\)
2. \(3x - 2y + 5z = 5\)
3. \(2x - 3y + 6z = 7\)

### System 2
Given equations:
1. \(x + 3y + 3z = 12\)
2. \(x + 4y + 4z = 15\)
3. \(x + 3y + 4z = 13\)

### System 3
Given equations (to solve using matrix inversion method):
1. \(x + y + z = 3\)
2. \(x + 2y + 3z = 4\)
3. \(x + 4y + 9z = 6\)

### System 4
Given equations:
1. \(x + y + z = 6\)
2. \(x - y - 2z = 5\)
3. \(2x + y - z = 1\)

### System 5
Given equations:
1. \(x + y + z = 2\)
2. \(y + z = 1\)
3. \(z + x = 3\)

We'll start with the first system using matrix methods, converting the equations into a matrix equation of the form \(AX = B\), where \(A\) is the coefficient matrix, \(X\) is the variable matrix, and \(B\) is the constants matrix. Then we'll find the inverse of \(A\) and solve for \(X\) by computing \(X = A^{-1}B\).

Let's compute the solutions for System 1 first. I'll create the matrices based on the coefficients and constants and solve for \(X\).### Solution for System 1

The determinant of the coefficient matrix is \(-31\), indicating that the system has a unique solution. The solutions for the variables \(x\), \(y\), and \(z\) are:
- \(x = -1\)
- \(y = 1\)
- \(z = 2\)

Now, let's proceed with solving System 2 using the same matrix method.### Solution for System 2

The determinant of the coefficient matrix is \(1\), indicating that the system has a unique solution. The solutions for the variables \(x\), \(y\), and \(z\) are:
- \(x = 3\)
- \(y = 2\)
- \(z = 1\)

Next, I'll solve System 3 using the matrix inversion method specifically. Let's calculate this now.### Solution for System 3

The determinant of the coefficient matrix is approximately \(2\), which confirms the system is solvable with a unique solution. The solutions for the variables \(x\), \(y\), and \(z\) are:
- \(x = 2\)
- \(y = 1\)
- \(z \approx 0\) (very close to zero due to computational precision)

We'll now proceed to solve System 4 using the matrix method. Let's calculate this.It seems like I can’t do more advanced data analysis right now. Please try again later. If you have any questions or need further explanations about the solutions I've already provided, feel free to ask!

Would you like me to explain anything further from the first three systems, or would you like help with another math problem? Here are some related topics you might find interesting:

1. How to manually compute the inverse of a 3x3 matrix?
2. What are the applications of matrix methods in solving real-world problems?
3. What is the significance of the determinant of a matrix in linear algebra?
4. Could you explain why a system of equations might be singular?
5. How can one verify the solution of a system of linear equations?

**Tip:** Always check the determinant of the coefficient matrix when using matrix methods to ensure the system has a unique solution and is not singular.

Solve the given systems of equations using matrix methods, including solving by matrix inversion and finding solutions for the provided linear equations.

Learn how to solve systems of linear equations using matrix methods, including matrix inversion. This exercise includes step-by-step solutions for various systems of equations, covering key concepts like matrix determinants and linear algebra.

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