Computing a kernel

System.
$$x+y=0, \qquad y+z=0, \qquad x+2y+z=0.$$
From the first equation, $x=-y$; from the second, $z=-y$. The third equation then reads $-y+2y-y=0$, which is automatically satisfied (it is the sum of the first two).

Kernel.
$$\ker f=\{(-y,\ y,\ -y):\ y\in\mathbb{R}\}=\operatorname{span}\big((-1,1,-1)\big).$$

Check. $f(-1,1,-1)=(0,\,0,\,-1+2-1)=(0,0,0)$.

Conclusion. $\ker f$ is the line spanned by $(-1,1,-1)$. In particular $f$ is not injective.