The plane x + y = 0

The correct option is "passes through z-axis".

To understand why the plane $x+y=0$ passes through the $z$-axis, let us analyze the equation of the plane and the definition of the $z$-axis.

First, we know that any point lying on the $z$-axis in three-dimensional space has coordinates of the form $(0,0,z)$, where $z$ can be any real number. This is because along the $z$-axis, both the $x$-coordinate and the $y$-coordinate are equal to zero.
Therefore, the equations representing the $z$-axis are:
$$x=0\text{and}y=0$$

Now, let us substitute the coordinates of any point on the $z$-axis, i.e., $x=0$ and $y=0$, into the equation of the given plane:
$$x+y=0$$
Substituting $x=0$ and $y=0$:
$$0+0=0$$

Since $0=0$ is a true statement, the coordinates of every point on the $z$-axis satisfy the equation of the plane. Consequently, the entire $z$-axis lies completely within the plane, which means the plane passes through the $z$-axis.