Answer
See solution
Work Step by Step
a) We build a system of three linear equations with three variables which has the solution $x=1$, $y=2$, $z=3$:
$$\begin{align*}
\begin{cases}
x+y+z&=1+2+3&&=6\\
2x-y+3z&=2(1)-2+3(3)&&=9\\
x-y+2z&=1-2+2(3)&&=5.
\end{cases}
\end{align*}$$
b) We build a system which has no solution: by doubling the left side of one equation without doubling the right side:
$$\begin{align*}
\begin{cases}
x+y+z&=6\\
2x-y+3z&=9\\
2x+2y+2z&=13.
\end{cases}
\end{align*}$$
c) We build a system which has infinitely many solutions: by doubling one equation:
$$\begin{align*}
\begin{cases}
x+y+z&=6\\
2x-y+3z&=9\\
2x+2y+2z&=12.
\end{cases}
\end{align*}$$
