Linear Equation Solver

A systematic algorithm to solve systems by transforming the augmented matrix to reduced row echelon form (RREF).

1. Forward elimination Use row operations to create zeros below each pivot
2. Scale pivots Divide each pivot row so the pivot becomes 1
3. Back substitution Eliminate above each pivot to get RREF

For a 2×2 system $Ax = b$, when $\det(A) \neq 0$:

Where $A_x$ replaces column 1 of $A$ with $b$, and similarly for $A_y$.

For a 2×2 matrix: $\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$

The rank of a matrix is the number of non-zero rows after row reduction (the number of pivot positions).