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
empty
a plane
one point
a line
1 Answer
one point.
Explanation:
The solutions would lie along a line if the 2 equations were consistent and linearly independent. The solution would be a single plane if the 2 equations were consistent but linearly dependent (this would result in you having only one equation).
The solution would be empty if the 2 equations were inconsistent ( this would result in the 2 planes being parallel). The solution being a point would only be true if you had 3 consistent linearly independent equations ( these would intersect to form a point, and therefore a unique solution).
Since you only have 2 equations a point can never be formed.
These ideas can be easily understood with the use of a 3D graph, but unfortunately this is not possible with the math tools on Socratic.
