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We all know that if we linearly combine two equations, say $f(x,y)=0$ and $g(x,y)=0$, and plot that ( $af(x,y)+bg(x,y)=0$ ), we find a graph that goes through the intersection points of $f(x,y)$ and $g(x,y)$, and why that happens is quite clear, as the points satisfies both equations should also satisfy the combined one.

So my question is what does the whole graph of the combined equation "visually" represent? In other sense, can you specify a point that lies on the "graph of combined equations" if "two points of two primary equations" with same $x$ value are given?

Bonnaduck
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N Kabir
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  • Visually, the larger $|a|$ is the more "influence" of $f$ on the shape of $af+bg=0$ has; similarly for $|b|$ and $g$. In general if $f(x_0,y_1)=0=g(x_0,y_2)$, then not much can be said about $y_0$ such that $af(x_0,y_0)+bg(x_0,y_0)=0$. E.g. consider $f(x,y)=x^2+y^2-1$, $g(x,y)=x-y$. Then $f(1,0)=0=g(1,1)$, but how many $y_0$'s such that $af(1,y_0)+bg(1,y_0)=0$ there are is highly dependent on $a$ and $b$. See Ex. 1 at https://www.desmos.com/calculator/vmbvs3vhc8 (and Ex. 2 for a more complicated example). – Alp Uzman Jan 04 '22 at 08:41

1 Answers1

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If we try to think about it on the $2$D plane, we may get no plausible answer. But if we view it from the perspective of a guy in the higher ($3$D) dimension (like Carl Sagan said), then it may become a better explanation.

Let's define two surfaces: $$S_1=\left(x,y,af(x,y)\right)\\S_2=\left(x,y,bg(x,y)\right)$$ Then the graph of $$af(x,y)-bg(x,y)=0$$ is a projection onto $xy$ plane of the cross-section of $S_1$ and $S_2$.

I drew how this looks like using the example from Alp Uzman. enter image description here enter image description here

Kay K.
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