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Elimination methods for systems of linear equations

• Purpose: find all solutions (x, y, z, …) to linear systems. Methods include substitution, elimination, and matrix-based approaches (Gaussian elimination and Gauss–Jordan elimination).

Substitution and elimination (two-equation example)

• Example system (from transcript):
  • x + 2y = 4
  • 3x + 4y = 10
• Elimination approach (combine equations to remove a variable):
  • Multiply the first equation by 3 to align x with the second: 3x + 6y = 12
  • Subtract the second equation from this result: (3x + 6y) - (3x + 4y) = 12 - 10 ⇒ 2y = 2 ⇒ y = 1
  • Substitute back into x + 2y = 4: x + 2(1) = 4 ⇒ x = 2
• Augmented-matrix representation (Gaussian elimination):
  • \begin{bmatrix}1 & 2 & | & 4 \ 3 & 4 & | & 10\end{bmatrix}
  • Row operation: R2 \leftarrow R2 - 3R_1 gives
    \begin{bmatrix}1 & 2 & | & 4 \ 0 & -2 & | & -2\end{bmatrix}
  • Solve: -2y = -2 \Rightarrow y = 1; then x = 2.
• Summary: Simple two-equation elimination yields a unique solution (x, y) = (2, 1).

Note: The transcript also shows other two-equation manipulations (e.g., replacing one equation by a linear combo) to illustrate that different sequences of eliminations lead to the same solution.

Gauss–Jordan elimination (row operations and augmented matrices)

• Core idea: transform the augmented matrix of a linear system to reduced row echelon form (RREF) using elementary row operations.
• Elementary row operations:
  • Swap two rows: Ri j
  • Multiply a row by a nonzero scalar: Ri ← cRi, c ≠ 0
  • Add a multiple of one row to another row: Ri ← Ri + cR_j, c ∈ R
• Example system (from transcript):
  • x + 2y + z = 8
  • x + y + z = 8
  • 3x + y − 2z = 9
• Gauss–Jordan steps (illustrative path to solution):
  • R2 ← R2 − R1 → [0, −1, 0 | 0]
  • R3 ← R3 − 3R1 → [0, −5, −5 | −15]
  • R2 ← −R2 → [0, 1, 0 | 0]
  • R3 ← R3 + 5R2 → [0, 0, −5 | −15]
  • R3 ← −R3/5 → [0, 0, 1 | 3]
  • R1 ← R1 − R3 → [1, 2, 0 | 5]
  • R1 ← R1 − 2R2 → [1, 0, 0 | 5]
• Final RREF gives the solution: x = 5,
  ewline y = 0,
  ewline z = 3.
• Takeaways:
  • Ga