Question

Consider the system of equations
\[
\begin{aligned}
&x+y-z=a \\
&x-y+2 z=b .
\end{aligned}
\]

a) Find the general solution of the homogeneous equation, so \(a=b=0\).

b) A particular solution of the inhomogeneous equations when \(a=1\) and \(b=2\) is \(x=1, y=1, z=1\). Find the most general solution of the inhomogeneous equations.

c) Find some particular solution of the inhomogeneous equations when \(a=-1\) and \(b=-2\).

d) Find some particular solution of the inhomogeneous equations when \(a=3\) and \(b=6\).

[Remark: After you have done part a), it is possible immediately to write the solutions to the remaining parts.]