A real system of three nondegenerate equations in three unknowns describes the common intersection of three planes. It can have no common point, one point, a line, or a plane of solutions. Degenerate equations $0=0$ or $0=c\ne0$ describe the whole ambient space or an impossible constraint instead of a plane.
For
$$x+y+z=6,\qquad x-y+z=2,\qquad x+y-z=0,$$
subtracting the second equation from the first gives $2y=4$, so $y=2$. Adding the first and third gives $2x+2y=6$, so $x=1$ and then $z=3$. The unique intersection is $(1,2,3)$, which satisfies all three equations.
In general, a consistent system's affine solution dimension is $3-\operatorname{rank}A$.
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