> [!summary] This note explains how to solve a linear system using parameterization in steps. The solution is a vector equation in terms of free variables as parameters. >[!info]+ Read Time **⏱ 2 mins** # Definition Solving a system of [[Linear Equations|linear equations]] using [[Parametrization|parametrization]] is done when the number of solutions is infinite. Meaning that a [[Free Variables|free variable]] is present. To solve a linear system in this way, the following steps must be done 1. Solve the [[Matrix Notation|augmented matrix]] into [[Reduced Row Echelon Form|reduced row echelon form ]] 2. Solve the [[Dependent Variables|dependent variables]] in terms of the [[Free Variables|free variables]] 3. Write the solutions as [[Vector Equations|a vector equation]] using the [[Free Variables|free variables]] as parameters # Examples > [!example] Solve a system of linear equations using parametrization > > $ \begin{array}{c} \\ \text{Step 1: Solve the augmented matrix (RREF)} \\ \\ \left[ \begin{array}{ccc|c} 1 & -2 & 1 & 3 \\ 2 & -4 & 3 & 9 \\ 3 & -6 & 2 & 6 \end{array} \right] \\ \\ R_2 \rightarrow R_2 - 2R_1, \quad R_3 \rightarrow R_3 - 3R_1 \\ \\ \left[ \begin{array}{ccc|c} 1 & -2 & 1 & 3 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & -1 & -3 \end{array} \right] \\ \\ R_3 \rightarrow R_3 + R_2 \\ \left[ \begin{array}{ccc|c} 1 & -2 & 1 & 3 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array} \right] \\ \\ R_1 \rightarrow R_1 - R_2 \\ \left[ \begin{array}{ccc|c} 1 & -2 & 0 & 0 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array} \right] \\ \\ \text{Step 2: Solve dependent variables in terms of free variables} \\ \\ x - 2y = 0 \Rightarrow x = 2y, \quad z = 3 \\ \\ \text{Let } y = t, \quad t \in \mathbb{R} \\ \\ \Rightarrow x = 2t, \quad y = t, \quad z = 3 \\ \\ \text{Step 3: Write the solution as a vector equation} \\ \\ \begin{bmatrix} x \\ y \\ z \end{bmatrix} = t \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ 3 \end{bmatrix}, \quad t \in \mathbb{R} \end{array} > $ # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/lb5tqk1EBeM?si=UphafFjay81jsV42" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>