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Your document is an assignment for a Linear Algebra course (MATH1003). The assignment contains several problems related to solving linear systems using methods like **Gauss-Jordan reduction**, determining the consistency of systems, and analyzing systems based on their augmented matrix form. Here's a summary of the problems:

1. **Determine Consistency and Solve Using Gauss-Jordan Reduction**:
   - Two systems of equations are given, and you're asked to determine if they are consistent and, if so, solve them using Gauss-Jordan reduction.
   
2. **Analyze the System Based on Parameters \( k \) and \( h \)**:
   - You need to find values or ranges of \( k \) and \( h \) such that the system has no solution, a unique solution, or infinitely many solutions.
   
3. **Solve a System Based on Augmented Matrix Form**:
   - For the given augmented matrix, determine the values of \( a \) and \( b \) for which the system has infinitely many solutions or is inconsistent.
   
4. **Solve Two Systems Using Matrix Operations**:
   - Two linear systems are given, and you're asked to incorporate the right-hand sides into a \( 2 \times 2 \) matrix and compute the reduced row echelon form of the augmented matrix.

Would you like help solving any of these problems in detail?

Here are 5 related questions to explore further:
1. How is the Gauss-Jordan reduction method applied to solve linear systems?
2. What conditions make a system of equations consistent or inconsistent?
3. How do parameter values like \( k \) and \( h \) affect the solutions of a system?
4. How can you determine if a system has infinitely many solutions from its augmented matrix?
5. What is the significance of the reduced row echelon form in solving linear systems?

**Tip:** When using Gauss-Jordan reduction, focus on getting leading 1's (pivot positions) in each row to make back substitution easier.

Indicate whether each of the following systems is consistent, and if it is consistent, use Gauss-Jordan reduction to solve the system.

Learn how to determine the consistency of linear systems and solve them using the Gauss-Jordan reduction method. Explore examples of consistent, inconsistent, and systems with infinitely many solutions, focusing on augmented matrices. Suitable for undergraduate Linear Algebra students.

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