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

The system cannot be ONE POINT.

Explanation:

Suppose you have 2 planes with the equations :

Plane `P`, with equation `Ax+By+Cz-D=0`

Plane `P_1`, with equation `A_1x+B_1y+C_1z-D_1=0`

The planes are parallel iif

`vecn=((A),(B),(C))` and `vecn'=((A_1),(B_1),(C_1))`

are collinear, that is

`vecn=kvecn'` where `k in RR`

If, `vecn` and `vecn'` are coplanar, then their intersection is either a line or empty

The only thing is that 2 planes cannor intersect in a point.