3x3 System with No Solution

Question

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?

Answer 1

A solution must satisfy all three equations. Plug $y=-1$ into the first and second equations and see what you get.

Answer 2

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.

Answer 3

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.

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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.)