[Math] For which values of $a, b$ does the system of equations NOT have any solutions

Question

I am trying to solve this math problem:

For which values of $a$ and $b$ does the linear system represented by the augmented matrix not have any solution?

$$ \left[
\begin{array}{ccc|c}
1&-2&3&-4\\
2&1&1&2\\
1&a&2&-b
\end{array}
\right] $$

Truthfully, I don't know where to start, so any help is welcomed.

Thank you very much!

EDIT: I tried to solve the problem on my one using Gaussian elimination, but I am not sure that the solution is the right one, or is that the right way to go.

EDIT2: OK, here is how I tried to solve it using Gaussian elimination:
multiplication of the third row with -1, and adding with the first row, the I got
$$ \left[
\begin{array}{ccc|c}
1&-2&3&-4\\
2&1&1&2\\
0&2a&-2&2b
\end{array}
\right] $$

Dividing the third row with 2:
$$ \left[
\begin{array}{ccc|c}
1&-2&3&-4\\
2&1&1&2\\
0&a&-1&b
\end{array}
\right] $$

Now my problem is, that I have two variables in one equation. I have no idea how to go next.

Answer 1

After a bit of Gaussian elimination, we arrive at

$$\begin{bmatrix}1 & 0 & 1 & 0 \\ 0 & 1 & -1 & 2\\ 0 & 0 & a+1 & -b - 2a\\ \end{bmatrix}$$

Now, when $a = - 1$ and $-b-2a \neq 0 \iff b\neq 2$, there will be no solution to the system of equations. Why not?