> [!summary] A free variable is a variable in a reduced linear system that does not have its own leading 1. > **Key Equations:** > Free variable defined as: $n\in \mathbb{R}$ >[!info]+ Read Time **⏱ 1 min** # Definition A free variable is a variable in a system of [[Linear Equations|linear equations]] that, after row reduction, does not have its own [[Leading 1s|leading 1]]. Since a free variable does not have its own constraints, any value can be assigned to it. If $n$ is a free variable, it can represent any [[Real Numbers|real number]], formally defined below $ n \in \mathbb{R} $ # Examples > [!example] Finding a free variable in a reduced linear system > Given the augmented matrix in [[Reduced Row Echelon Form|reduced row echelon form]]: > $ \left[ \begin{array}{ccc|c} 1 & 0 & \color{red}{2} & 3 \\ 0 & 1 & \color{red}{-1} & 4 \\ 0 & 0 & \color{red}{0} & 0 \\ \end{array} \right] > $ The red entries are a free variable since they do not have their own leading 1 in a reduced row echelon form. --- > 🧪 Think this could help someone else? [Star Math & Matter on Github](https://github.com/rajeevphysics/Obsidian-MathMatter) to help more learners discover it. ---