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?

Question

empty
a plane
one point
a line

1265 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-09-22T15:16:35

The system cannot be ONE POINT.

Suppose you have 2 planes with the equations :

Plane $P$ $P$ , with equation $A x + B y + C z - D = 0$ $Ax+By+Cz-D=0$

Plane $P_{1}$ $P_1$ , with equation $A_{1} x + B_{1} y + C_{1} z - D_{1} = 0$ $A_1x+B_1y+C_1z-D_1=0$

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.