Algebra → Linear Algebra · Lesson 6

Systems of Equations & Gaussian Elimination

Large systems of linear equations — 10, 100, 1000 unknowns — are solved by Gaussian elimination. This algorithm powers circuit analysis, physics simulations, and machine learning.

Key Concepts

Augmented Matrix

A system of equations can be written as an augmented matrix [A|b]. Example: 2x + y = 5, x − y = 1 becomes [[2,1,5],[1,-1,1]]. Row operations on the matrix equal operations on the equations.

Row Operations

Three legal operations (they don't change the solution): 1) Swap two rows. 2) Multiply a row by a nonzero constant. 3) Add a multiple of one row to another.

Row Echelon Form

Use row operations to put the matrix in a staircase shape: each row's leading entry (pivot) is to the right of the one above. Then back-substitute from the bottom up.

Reduced Row Echelon Form

RREF: each pivot is 1, and all other entries in the pivot column are 0. The Gauss-Jordan method produces RREF directly, making back-substitution unnecessary.

Live Python Practice

Interactive Lab

Row 1: x + y =
Row 2: x + y =

Watch Gaussian elimination unfold step by step. The green column is the right-hand side (constants).

Check Your Understanding

An augmented matrix [A|b] represents:

Which is a valid row operation?

In row echelon form, the leading entry of each row is:

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