I'm having trouble figuring out how to get the "generic solution" to this matrix by using Gaussian Method by REF. The answer on the back of the book is: $(x,y,\frac{1}{2}x-\frac{3}{2}y)$. I really don't understand the concept behind this answer so if anyone can help, I'd really appreciate it! thank you. I've attached a photo of the problem.
$$2x+3y+2z=1$$
and there is no other constraints as $0=0$ trivially holds.
Then $$z=\frac{1-3y-2x}{2}=\frac{1}{2}-x-\frac{3y}{2}$$
Hence, given $x,y$, the $z$ must satisfies $z=\frac{1-3y-2x}{2}=\frac{1}{2}-x-\frac{3y}{2}.$