3x3 System with No Solution I have the system of equations as follows:
$x + y + z =0$
$2x + 3y + 2z = -3$
$-x - 2y -z = 1$
First I added the first and third equation and the result was that $y = -1$.
Next I plugged $y = -1$ into the first and third equation. The result was the true statement $1 = 1$.  The book says this problem has no solutions but I thought that when you get a true statement it means there are infinite solutions.  Or is there no solution because $y=-1$ and the other two resulted in infinite solutions?
 A: A solution must satisfy all three equations.  Plug $y=-1$ into the first and second equations and see what you get.
A: Note that one could double the first equation and subtract it from the first equation to produce the result of $y=-3$ which would be in conflict with the $y=1$ of how the first and third planes intersect. Thus there is no solution for all the equations.
Here's a PDF with more of an explanation if you need it.
A: You wrote:

Next I plugged $y = -1$ into the first and third equation. The result was the true statement $1 = 1$. 

I am not sure how exactly you get $1=1$. But when plugging $y=-1$ into these three equations I get:
$\begin{align*}
x -1 + z &=0\\
2x -3 + 2z &= -3\\
-x +2 -z &= 1
\end{align*}$
which is the same as
$\begin{align*}
x + z &=1\\
2x  + 2z &= 0\\
-x  -z &= -1
\end{align*}$
It should be relatively easy to see, that this system has no solution.

If you did not use the the second equation at all, that would mean the you were solving the system of these two equations:
$\begin{align*}
x + y + z &=0\\
-x -2y -z &= 1
\end{align*}$
This system has solutions. (For example, $x=1$, $y=-1$, $z=0$ is one of the solutions.)
