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

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Answer 1 · 2017-09-22T15:16:35

The system cannot be ONE POINT.

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.