$$ \left[ \begin{array}{cc|c} 3&4&5\\ 4&5&4 \end{array} \right] $$
Could I say that this augmented matrix forms two different planes ($3x_1 + 4x_2 = 5$ and $4x_1 + 5x_2 = 4$)? And the vector $(x_1,x_2)$ is the vector intersection of those two planes?
Since there are only two rows in the matrix and two columns without the augment, we could say that the augmented matrix represents two different lines or two simultaneous linear equations in two unknowns, and the solution of the equations is the intersection point of those two lines. There is no need for a plane other than the Euclidean one of all ordered pairs of real numbers.