I have that
$A_1x + B_1y + C_1z = D_1$
$A_2x + B_2y + C_2z = D_2$
How do I intepret the solutions to these equations geometrically? What do they mean?
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In $\mathbb R^3$, the solution set of a single linear equation is a plane (not necessarily through the origin). The solution set of a system of equations is the intersection of the solution sets of the equations in the system.
So...the solution set of a system of two linear equations is the intersection of two planes. This is either a plane (when the two equations describe the same plane), a line (if they are not the same plane and not parallel), or empty (if they are parallel).
There's a couple different ways.
Each equation can be considered a two dimensional plane in three dimensional space. The solutions $(x,y,z)$ to the equation are the points of the plane. Simultaneous solutions to the two equations will be where the planes intersect. Note this will generally happen in a line, but it could also not happen at all (if the planes are parallel and separated) or happen in a plane (if the planes coincide).
The $(A_1,A_2)$, $(B_1,B_2),$ $(C_1,C_2)$ and $(D_1,D_2)$ can be seen as two dimensional vectors and the two equations can be considered as $xA+yB+zC=D$ which says that $D$ is a linear combination of vectors $A,$ $B$ and $C.$ So the solutions are different ways to make the vector $D$ as a linear combination of $A$ $B$ and $C.$ Note that since the vectors are two-dimensional, as long as two of $A,B,C$ are linearly independent, you can always make $D$ as a linear combination(i.e. there is a solution). In this case, the third vector (if nonzero) will be redundant and this corresponds to the generic case above where the solution set forms a line (the one redundant vector is the one degree of freedom of the line). In the weird case where $A$ $B$ and $C$ are all parallel to one another and not linearly independent, then you can only make $D$ as a linear combination if it is parallel to them, and otherwise you can't (i.e. no solution). The first case here corresponds to the case above where the solution set forms a plane and the second to when there are no solutions (when the solutions to the individual equations were parallel, separated planes).