So, we have a system of linear equations.It's preferred that we visualize the column picture, since as dimensions go up we only have to think of vectors moving to more dimensions, instead of hyperplanes. So, given the following equations, how is it that either no matter if we view them as lines or vectors, the solution is the same? $$ \begin{cases}2x + 4y = 9\\ 7x - 12y = 11 \end{cases} $$
Hopefully my question is not too confusing.
The hyperplane (affine subspace of dimension $n–1$) in 2D is a line (affine subspace of dimension $1$).
So each of your linear equations, $$ a_1 x_1 + \dotsb + a_n x_n = b $$ which in general each is the equation of a hyperplane, $$ a \cdot x = b $$ with normal vector $a$ and distance $b/\lVert a \rVert$ to the origin, for two dimensions, is the equation of a line as well. $$ a_1 x_1 + a_2 x_2 = b $$