Question

In general, a linear equation in three variables x,y,z has the form Ax+By+Cz=D (where A,B,C,D are some constants) and geometrically corresponds to a plane in 3D. Graphically, the solution of a system of two linear equations in three variables CANNOT be?

empty
a plane
one point
a line

Answer 1

A point.

Explanation:

The graphical "solution" is the intersection of the two planes. A "solution" would imply that it isn't "empty" to me, but that would be a possible outcome of the equations. It could also be a plane, if the two functions are in fact identical. Most commonly, a realistic solution is the line defined by the intersection of one plane through another.

Answer 2

a plane

Explanation:

The solution of two lines in `RR^3`

`L_1 -> p=p_1+lambda_1 vec v_1`
`L_2->p=p_2+lambda_2 vec v_2`

is given by the solutions for

`p=p_1+lambda_1 vec v_1=p_2+lambda_2 vec v_2`

Also `L_1, L_2` can be coincident for instance when

`p_2 = p_1 + lambda_1 vec v_2` and `vec v_2 = mu vec v_1`

and also they can be non intersecting.

The answer is a plane