So I'm doing linear algebra right now and I have a question regarding addition of equations as part of Gauss' elimination algorithm. I understand why it's possible, as the LHS of one equation can be added/subtracted to another equation as long as its RHS is also added/subtracted. But I'm trying to better understand about the intersecting point of the equations.
Given two equations: $$x + 2y = 3$$ $$3x + 4y = 5$$
they intersect at point (-1, 2). When I add other equations which are combinations of these two equations (ie $-x - y = -1$), they also all pass through this point. Maybe I'm missing something, but could someone point out which property allows for this? Why does a new equation, composed of some combination of two others, resulting in an equation with a different slope and intercepts, still pass through that same point? What is so special about that point?
I'm a pretty visual learner, so any visual analogies would be much appreciated. Thanks for your time!