https://file.aiquickdraw.com/mathbot/file/17363483102119j98wuyt.jpg

The image contains two sets of systems of linear equations to solve using the matrix method. Here's the breakdown of the given problems:

### System (i):
The equations are:
1. \( x - y + z = 4 \)
2. \( 2x + y - 3z = 0 \)
3. \( x + y + z = 2 \)

### System (ii):
The equations are:
1. \( x + y + z = 2 \)
2. \( 2x + 2y + 3z = 7 \)
3. \( 5y - z + 13 = 0 \)

I will solve **both systems** using the matrix method step by step. Let me calculate them for you. numpy as np

# System (i)
A1 = np.array([[1, -1, 1],
               [2,  1, -3],
               [1,  1, 1]])
B1 = np.array([4, 0, 2])

# Solve for (i)
solution_i = np.linalg.solve(A1, B1)

# System (ii)
A2 = np.array([[1, 1, 1],
               [2, 2, 3],
               [0, 5, -1]])
B2 = np.array([2, 7, -13])

# Solve for (ii)
solution_ii = np.linalg.solve(A2, B2)

solution_i, solution_iiHere are the solutions to the systems of equations:

### For System (i):
The solution is:
\[
x = 2, \; y = -1, \; z = 1
\]

### For System (ii):
The solution is:
\[
x = 1, \; y = -2, \; z = 3
\]

Let me know if you'd like detailed steps for solving these using the matrix method or have further questions!  

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Here are 5 follow-up questions related to this topic:  
1. How do you write a system of linear equations in matrix form?  
2. What is the determinant, and how does it relate to solving a system of equations?  
3. What does it mean if the determinant of the coefficient matrix is zero?  
4. Can all systems of linear equations be solved using the matrix method? Why or why not?  
5. How do you interpret the solution of a linear system geometrically?

**Tip:** When solving linear equations with the matrix method, always ensure the coefficient matrix is invertible (non-zero determinant) for a unique solution!

Solve the given systems of linear equations using the matrix method:
(i) x - y + z = 4, 2x + y - 3z = 0, x + y + z = 2
(ii) x + y + z = 2, 2x + 2y + 3z = 7, 5y - z + 13 = 0

Learn how to solve systems of linear equations using the matrix method. Example systems include equations such as x - y + z = 4, 2x + y - 3z = 0, and more. Discover step-by-step solutions involving linear algebra concepts like matrix equations and determinants. Suitable for high school students in Grades 10-12.

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