How are theses functions called and what are their properties? We define a system of two equations with two variables and six parameters:
$\begin{cases}
a_1x+b_1y=c_1\\
a_2x+b_2y=c_2
\end{cases}$
We obtain a function $f:\mathbb{R^3\times\mathbb{R}^2\rightarrow\mathbb{R}^2}$,
$$\pmatrix{a_1 &  a_2 \\ b_1 & b_2 \\ c_1 & c_2}\mapsto \pmatrix{x \\ y} $$
If we fix the five parameters $a_1$, $a_2$, $b_1$, $b_2$, $c_1$ and we vary the value of $c_2$. We obtain a function $f_{c_2}: \mathbb{R}\rightarrow\mathbb{R}^2$,
$c_2\mapsto\left(x,y\right)$. In the same manner we obtain the five other functions $f_{a_1}$, $f_{a_2}$, $f_{b_1}$, $f_{b_2}$, $f_{c_1}$.
My question is how are theses functions called and what are their properties?
 A: The functions in OP are defined by the system:
$$\begin{cases}
a_1x+b_1y=c_1\\
a_2x+b_2y=c_2
\end{cases}$$
so that:
$$
x=\dfrac {\Delta_x}{\Delta} \qquad y=\dfrac {\Delta_y}{\Delta}
$$
where:
$$
\Delta=a_1b_2-a_2b_1 \quad \Delta_x=b_1c_2-b_2c_1 \quad \Delta_y=a_1c_2-a_2c_1
$$
so that the functions $f_{c_1}:\mathbb{R}\to\mathbb{R}^2$ and $f_{c_2}:\mathbb{R}\to\mathbb{R}^2$ are:
$$
f_{c_1}(t)=
\begin{bmatrix}
\frac{-b_2}{\Delta}t+\frac{b_1c_2}{\Delta}\\
\frac{-a_2}{\Delta}t+\frac{a_1c_2}{\Delta}\\
\end{bmatrix}
$$
and
$$
f_{c_2}(t)=
\begin{bmatrix}
\frac{b_1}{\Delta}t-\frac{b_2c_1}{\Delta}\\
\frac{a_1}{\Delta}t-\frac{a_2c_1}{\Delta}\\
\end{bmatrix}
$$
so these functions represent straight lines on $\mathbb{R}^2$.
The functions $f_{a_i}$ and $f_{b_i}$ are different. As example the function $f_{a_1}:\mathbb{R} \to \mathbb{R}^2$ is:
$$
f_{a_1}(t)=
\begin{bmatrix}
\dfrac{b_1c_2-b_2c_1}{b_2t-a_2b_1}\\
\dfrac{c_2t-a_2c_1}{b_2t-a_2b_1}
\end{bmatrix}
$$
that represent an homographic function in $\mathbb{R}^2$ whose graph is an hyperbola.
I dont know if all these function have some collective name (I never see it).
1.
