Basic and free variables

In a solution to a system of linear equations,

Free variables (also free parameters) refer to the variables that are undetermined and may take any value. In row echelon form they correspond to a column that does not contain a leading entry.
Basic variables (also leading variables) refer to the variables that are determined as combinations of free variables and constants. These correspond to columns that contain a leading entry.

Note that the selection of which variables are basic and which are free is arbitrary, depending only on how the solution is written. For example, $𝑥 = 2 𝑦$ , where $𝑥$ is basic and $𝑦$ is free, may equivalently be expressed as $𝑦 = 12 𝑥$ , where $𝑦$ is basic and $𝑥$ is free.

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