Geometric interpretation of the addition of linear equations in general form I have a very simple question: suppose I have two 2D linear equations in general form
$$ a_1x + b_1y + c_1 = 0$$
$$ a_2x + b_2y + c_2 = 0$$
I'd like to know what's the (intuitive) geometric interpretation of their addition and subtraction
$$ (a_1 + a_2)x + (b_1 + b_2)y + (c_1 + c_2) = 0$$
$$ (a_1 - a_2)x + (b_1 - b_2)y + (c_1 - c_2) = 0$$

Note that you can write the linear equations above as follows:
$$
\begin{pmatrix}
a_1& b_1\\
a_2&b_2 
\end{pmatrix}
\begin{pmatrix}
x\\
y 
\end{pmatrix}=
\begin{pmatrix}
-c_1\\
-c_2
\end{pmatrix}
$$
Now the addition and subtraction means multiplication of the previous equation by following matrix:
$$
T=
\begin{pmatrix}
1& -1\\
1& 1 
\end{pmatrix}=\sqrt 2 
\begin{pmatrix}
\cos(\frac{\pi}{4})&-\sin(\frac{\pi}{4})\\
\sin(\frac{\pi}{4})&\cos(\frac{\pi}{4}) 
\end{pmatrix}=\sqrt 2 R_{\frac{\pi}{4}}.
$$
$R_{\frac{\pi}{4}}$ is the rotation matrix and therefore the effect of multiplication by $T$ is indeed the rotation of lines by $\frac{\pi}{4}$ and then the scaling (homothety) by $\sqrt 2$.
