📘 Chapter 8: Systems of Linear Equations

🔍 What is a System of Equations?

A system of linear equations is a set of equations involving the same variables. Example:

🧠 Methods to Solve

• Substitution & Elimination (small systems)
• Matrix Representation: \( AX = B \)
• Gaussian Elimination (Row-echelon form)
• Gauss–Jordan Elimination (Reduced row-echelon form)
• Matrix Inverse Method: \( X = A^{-1}B \) (if \( A \) is invertible)

📊 Types of Solutions

• Unique Solution: Exactly one solution
• Infinitely Many Solutions: Depends on free variables
• No Solution: Inconsistent system

📌 Example

Use Gaussian elimination on: \[ \begin{aligned} x + y + z &= 6 \\ x + 2y + 3z &= 14 \\ 2x + y + z &= 10 \end{aligned} \] Convert to augmented matrix and apply row operations to solve.

📝 Practice

Q: Solve the following system using any method: \[ \begin{aligned} 2x + 3y &= 8 \\ 4x + 9y &= 20 \end{aligned} \]