When a column has no pivot, its unknown is free: it can take any value, and the other unknowns follow from it. Each free unknown is a direction along which the solutions stretch, so one gives a line of solutions and two give a plane.

A recipe for a blend fixes some amounts once others are chosen. The free ones are the choices you are allowed; everything else is forced by them.

The solutions of $x+2y=4$, one equation in two unknowns. With $y=t$ free, $x=4-2t$, so every solution is $\mathbf{p}+t\mathbf{v}$ with $\mathbf{p}=(4,0)$ and $\mathbf{v}=(-2,1)$: the point $\mathbf{p}$ moved along the direction $\mathbf{v}$. One free unknown, one line.

After elimination, an unknown whose column holds a pivot is a basic variable, and every other unknown is free. If the system is consistent, the free variables may take any values, and back substitution expresses each basic variable through them. Collected into one vector, the result is the parametric form of the solutions: a particular solution plus a combination of fixed vectors, one for each free variable.

• A row $(0\ \cdots\ 0\mid c)$ with $c\neq0$ is the equation $0=c$: the system has no solution.

Subtract the two equations: $A\mathbf{x}-A\mathbf{p}=\mathbf{b}-\mathbf{b}=\mathbf{0}$, and $A\mathbf{x}-A\mathbf{p}=A(\mathbf{x}-\mathbf{p})$. So every solution is plus a solution of the homogeneous system, and every such sum is a solution.

Fix one particular solution $\mathbf{p}$.

A matrix distributes over a difference of vectors.

If $\mathbf{x}$ is a solution too, the difference solves the homogeneous system: $\mathbf{b}-\mathbf{b}=\mathbf{0}$.

Conversely, $\mathbf{p}$ plus any homogeneous solution solves the system.

The solution set is one particular solution moved by every solution of the homogeneous system.

Look at the echelon form. Unknowns whose columns hold pivots are basic; the rest are free.

Pivots in columns 1 and 3: $x_1$ and $x_3$ are basic, $x_2$ and are free. Then and .

• Pivot columns give basic variables; the other columns give free variables.
• A row $(0\ \cdots\ 0\mid c)$ with $c\neq0$ means no solution.
• Consistent with no free variable: one solution. With one or more: infinitely many.
• Every solution is a particular one plus a solution of .

Great work! You now know how to:

• tell basic variables from free ones;
• write the solutions in parametric form;
• decide between no solution, one, and infinitely many;
• describe every solution as a particular one plus a homogeneous one.