We will describe the solution space of the equation
\begin{equation*}
\left[\begin{array}{rrr}
2 \amp 0 \amp 2 \\
4 \amp -1 \amp 6 \\
1 \amp 3 \amp -5 \\
\end{array}\right]
\xvec
=
\left[\begin{array}{r}
0 \\ -5 \\ 15
\end{array}\right].
\end{equation*}
This equation may be equivalently expressed as
\begin{equation*}
x_1\left[\begin{array}{r}2\\4\\1\end{array}\right] +
x_2\left[\begin{array}{r}0\\-1\\3\end{array}\right]+
x_3\left[\begin{array}{r}2\\6\\-5\end{array}\right]=
\left[\begin{array}{r}0\\-5\\15\end{array}\right]\text{,}
\end{equation*}
which is the linear system corresponding to the augmented matrix
\begin{equation*}
\left[\begin{array}{rrr|r}
2 \amp 0 \amp 2 \amp 0 \\
4 \amp -1 \amp 6 \amp -5 \\
1 \amp 3 \amp -5 \amp 15 \\
\end{array} \right]\text{.}
\end{equation*}
The reduced row echelon form of the augmented matrix is
\begin{equation*}
\left[\begin{array}{rrr|r}
2 \amp 0 \amp 2 \amp 0 \\
4 \amp -1 \amp 6 \amp -5 \\
1 \amp 3 \amp -5 \amp 15 \\
\end{array} \right]
\sim
\left[\begin{array}{rrr|r}
1 \amp 0 \amp 1 \amp 0 \\
0 \amp 1 \amp -2 \amp 5 \\
0 \amp 0 \amp 0 \amp 0 \\
\end{array} \right],
\end{equation*}
which corresponds to the linear system
\begin{equation*}
\begin{alignedat}{4}
x_1 \amp \amp \amp {}+{} \amp x_3 \amp {}={} \amp 0 \\
\amp \amp x_2 \amp {}-{} \amp 2x_3 \amp {}={} \amp 5. \\
\end{alignedat}
\end{equation*}
The variable \(x_3\) is free so we may write the solution space parametrically as
\begin{equation*}
\begin{aligned}
x_1 \amp {}={} -x_3 \\
x_2 \amp {}={} 5+2x_3. \\
\end{aligned}
\end{equation*}
Since we originally asked to describe the solutions to the equation \(A\xvec = \bvec\text{,}\) we will express the solution in terms of the vector \(\xvec\text{:}\)
\begin{equation*}
\xvec
=\left[
\begin{array}{r}
x_1 \\ x_2 \\ x_3
\end{array}
\right]
=
\left[
\begin{array}{r}
-x_3 \\ 5 + 2x_3 \\ x_3
\end{array}
\right]
=\left[\begin{array}{r}0\\5\\0\end{array}\right]
+x_3\left[\begin{array}{r}-1\\2\\1\end{array}\right]
\end{equation*}
As before, we call this a parametric description of the solution space.
This shows that the solutions \(\xvec\) may be written in the form \(\vvec + x_3\wvec\text{,}\) for appropriate vectors \(\vvec\) and \(\wvec\text{.}\) Geometrically, the solution space is a line in \(\real^3\) through \(\vvec\) moving parallel to \(\wvec\text{.}\)
