Write the general solution to the following linear systems in the form (2.27). Clearly identify the particular solution $\mathbf{x}^{\star}$ and the element $\mathbf{z}$ of the kernel. (a) $x-y+3 z=1$,
(b) $\left(\begin{array}{rrr}1 & -2 & 0 \\ 2 & 3 & 1\end{array}\right)\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{r}3 \\ -1\end{array}\right)$,
(c) $\left(\begin{array}{rrr}1 & -1 & 0 \\ 2 & 0 & -4 \\ 2 & -1 & -2\end{array}\right)\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{l}-1 \\ -6 \\ -4\end{array}\right)$,
(d) $\left(\begin{array}{rrr}2 & -1 & 1 \\ 4 & -1 & 2 \\ 0 & 1 & 3\end{array}\right)\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{r}0 \\ 1 \\ -1\end{array}\right)$,
(e) $\left(\begin{array}{rr}1 & -2 \\ 2 & -4 \\ -3 & 6 \\ -1 & 2\end{array}\right)\left(\begin{array}{l}u \\ v\end{array}\right)=\left(\begin{array}{r}-1 \\ -2 \\ 3 \\ 1\end{array}\right)$,
(f) $\left(\begin{array}{rrrr}1 & -3 & 2 & 0 \\ -1 & 5 & 1 & 1 \\ 2 & -8 & 1 & -1\end{array}\right)\left(\begin{array}{l}p \\ q \\ r \\ s\end{array}\right)=\left(\begin{array}{r}4 \\ -3 \\ 7\end{array}\right)$,
(g)
$$
\left(\begin{array}{rrrr}
0 & -1 & 2 & -1 \\
1 & -3 & 0 & 1 \\
-2 & 5 & 2 & -3 \\
1 & 1 & -8 & 5
\end{array}\right)\left(\begin{array}{c}
x \\
y \\
z \\
w
\end{array}\right)=\left(\begin{array}{r}
-2 \\
-3 \\
4 \\
5
\end{array}\right)
$$