Parametric vector form of the solution set of a system of linear equations

Definition: Gaussian elimination is a systematic method for solving systems of linear equations by transforming the augmented matrix $[A | b]$ into echelon form using three types of elementary row operations:

1. Swap two rows: $R_i \leftrightarrow R_j$
2. Scale a row by a nonzero constant: $R_i \leftarrow c \cdot R_i$
3. Replace a row by itself plus a multiple of another: $R_i \leftarrow R_i + c \cdot R_j$

Algorithm:

1. Write the system as an augmented matrix
2. Use row operations to create zeros below each pivot (left to right, top to bottom)
3. The resulting echelon form reveals the solution by back-substitution

Example:
$$\begin{bmatrix} 2 & 1 & | & 5 \\ 1 & -3 & | & -1 \end{bmatrix} \xrightarrow{R_1 \leftrightarrow R_2} \begin{bmatrix} 1 & -3 & | & -1 \\ 2 & 1 & | & 5 \end{bmatrix} \xrightarrow{R_2 - 2R_1} \begin{bmatrix} 1 & -3 & | & -1 \\ 0 & 7 & | & 7 \end{bmatrix}$$

From the echelon form: $y = 1$, $x = -1 + 3(1) = 2$.