Concept clarification in math

There’re been quite a few aspects of mathematical object I’m come across anew, even more in a lecture as when I found difficulties in simplifying the context.

During my math lecture today, I struggled to explain the reason why

kf(x,y) + g(x,y) = 0

denotes a line passes through the intersection point of non-paralleled two lines provided for as f(x,y)=0 and g(x,y)=0; whereas k, x, y are in $latex \mathbb{R}$.

I believed there were at least algebraic and analytic approaches.

In the words of algebra, it could be thought that a line passing through a fixed point P=P(a,b) is specified by single given point so that the whole lines in such condition form one dimension vector space. Besides, the P-passing line can be represented as t(x-P)=0 (s.t. x∈R^2, t∈R). Now that let h(x,y) be a linear function as in h(x,y)=kf(x,y)+g(x,y), then h(P)=kf(P)+g(P)=0. According to the Parallel Postulate of Euclid’, there uniquely exists the intersection point of (f(x,y)=0) and (g(x,y)=0), which only rephrases that when f(x,y) and g(x,y) become 0 at P simultaneously, then that is the point they intersect.

In this argument, “the P-passing attribute” of a straight line L:h(x,y)=0 shall be mathematically reworded as “P is the root of polynomial h(x,y)”, and vice versa. On an occasion, I want to know a polite way of explaining them by the vector equation.