How does existence of pivots affect the solution set

• Since column 2 has no pivot, $x_2$ does not have an expression of its own.

• Instead $x_2$ is used inside $x_1$‘s expression.

• Since there are no restrictions on $x_2$, we call it a free parameter.

• To make the freeness clear, we replace $x_2$ with $t$ in the equations.

• So if a column doesn’t have a pivot, we get a free variable.