My teacher said that whenever you're given a line, say $ax+by+c=0$,
and a linear relation in $a$, $b$ and $c$, say $2a + 3b+c=0$,
these two sets of equations represent a family of lines {in this case passing through $(2,3)$}.
However, he did not elaborate on it. The books that I have at home only mentions the form $L_1 + k L_2$ = 0, where $L_1$ and $L_2$ are two non-parallel lines and $k$ is a parameter. I know that when we eliminate $c$ from the equation of the line by substituting its value from the linear relation, we will get the family of lines in the form $a(x-2) +b (y-3) = 0$ and $k =a/b$.
But can someone intuitively explain why the former happens... (like in the case of the latter $L_1$ and $L_2$ both satisfy that point, so $0 + k 0 = 0 $, something like this which helps me get the 'feel' for it). P.S. Sorry I couldn't find a better word for feel.
Observe that $2a+3b+c=0$ is the equation that results from substituting the coordinates of the point $(2,3)$ into the equation $ax+by+c=0$ of a line. So, if you have three numbers $a$, $b$ and $c$ that satisfy this linear constraint, then the equation of a line that has those three numbers for its coefficients is satisfied by $(2,3)$. To put it another way, the linear relation $2a+3b+c=0$ expresses the fact that $(2,3)$ lies on the line $ax+by+c=0$.