Why is it sometimes possible to solve multi variable single equation? Assume you have an equation:
$\ 5x^2+4x+6=ax^2+bx+c $
Now theoretically, this is an equation of 4 variables, and it should not be solvable, but it is very apparent that a,b,c equals 5,4,6, in that order. Something similar happens with complex numbers:
$\ 5 + 4i = a + bi $
Here solution for a,b is 5,4, again it is very apparent.
My theory is that a,b or c, in these are something like incompatible number types. Therefore you can construct a system of equations from one single equation. The complex number equation is a good example, you can easily separate the complex and the non-complex part, as they virtually cannot influence the other. 
But in the first equation, it's all in the real plane, there is no complex stuff going on and the x squared can influence the x
You can even do this:
$\ c = 5x^2 + 4x + 6 - ax^2 - bx $
Which would suggest c is dependent on the value of both a and b, and you get a similar result defining a or b, potentially pointing to an infinite number of solutions.
So why do these have only one solution? Assuming my theory is somewhat correct, what do mathematicians call these "incompatible numbers" properly?

Your question touches on a very important concept in linear algebra, that of linear independence. Some set of vectors $v_1,v_2,\dots,v_n$ are linearly independent if the only solution to $a_1v_1+a_2v_2+\dots+a_nv_n=0$ is when $a_1=a_2=\dots=a_n=0$. For example, $x^2$, $x$, and $1$ are linearly independent because if $f(x)=ax^2+bx+c$ and for any $x$, $f(x)=0$ (ie. $f$ is the $0$ function) then $a=b=c=0$. Now, a very important concept in linear algebra is that if we have some vector $v=a_1v_1+a_2v_2+\dots+a_nv_n$ there is only one possible choice of $a_1,a_2,\dots,a_n$. That means that if $ax^2+bx+c=dx^2+ex+f$, $a=d$, $b=e$, and $c=f$. In addition, note that $1$ and $i$ are linearly independent over the reals because the only real numbers $a,b$ where $a(1)+b(i)=0$ are $a=b=0$, so this property holds again. If you are interested in this topic, I recomend you research linear independence and linear algebra.
