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
$$ (a_1 + a_2)x + (b_1 + b_2)y + (c_1 + c_2) = 0$$
$$ (a_1 - a_2)x + (b_1 - b_2)y + (c_1 - c_2) = 0$$

In general, if you have a system of two linear equations whose solution is a line $L$ in $3$-space, you can visualize the general linear combination of the equations as giving another plane containing $L$. Think of this as different positions of a revolving door, pivoting around $L$.
