Solve linear systems using Gaussian elimination, understand rank and solution classification, and interpret Ax = b as linear combinations. Master systematic methods for solving systems of any size.
Solve the system of linear equations!
System of Equations:
Classify the solution type of the system!
System:
Gaussian elimination uses elementary row operations to transform an augmented matrix [A|b] into row-echelon form, making it easy to solve the system by back substitution.
System: 2x + 3y = 7, x + 4y = 9
Step 1: Augmented matrix: [[2, 3 | 7], [1, 4 | 9]]
Step 2: Row operations to get: [[1, 0 | 1], [0, 1 | 2]]
Answer: x = 1, y = 2
System: x + y + z = 6, 2x - y + z = 3, x + 2y - z = 0
Step 1: Create augmented matrix
Step 2: Apply row operations to achieve row-echelon form
Step 3: Back substitute to find: x = 1, y = 2, z = 3
Answer: x = 1, y = 2, z = 3
Goal: Transform to [[1, 0, 0 | a], [0, 1, 0 | b], [0, 0, 1 | c]]
Advantage: Solutions can be read directly from RREF
Method: Continue row operations until pivots are 1 with zeros above and below