Find three different systems of linear equation whose solutions are.. Find three different systems of linear equation whose solutions are $x_1 = 3, x_2 = 0, x_3 = -1$
I'm confused, how exactly can I do this?

Note that your system is described by the matrix
$$
\left[\begin{array}{rrr|r}
1 & 0 & 0 & 3 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & -1
\end{array}\right]
$$
Performing any row operation on this matrix yields a system with the same solutions. For example, you could add $\DeclareMathOperator{Row}{Row}\Row_1$ to $\Row_2$ to get the system
$$
\left[\begin{array}{rrr|r}
1 & 0 & 0 & 3 \\
1 & 1 & 0 & 3 \\
0 & 0 & 1 & -1
\end{array}\right]
$$
You could then add $\Row_2$ to $\Row_3$ to get 
$$
\left[\begin{array}{rrr|r}
1 & 0 & 0 & 3 \\
1 & 1 & 0 & 3 \\
1 & 1 & 1 & 2
\end{array}\right]
$$
To get a third system, you could then add $2\cdot \Row_3$ to $\Row_1$
$$
\left[\begin{array}{rrr|r}
3 & 2 & 2 & 7 \\
1 & 1 & 0 & 3 \\
1 & 1 & 1 & 2
\end{array}\right]
$$
