Homogeneous linear equations geometrically

[phymatter]

what does a homogeneous linear equation in 3 variables represent geometrically ?

 

[!^--_--^!]

A plane

 

[HallsofIvy]

A homogeneous linear equation is of the form ax+ by+ cz= 0 for numbers a, b, c, d. If all of a, b, c, d are 0, that is true for all (x, y, z) so the solution set is all space.

If any of a, b, c, is not 0, we can solve for that variable:
x= (-by- cz)/a
y= (-ax- cz)/b
z= (-ax- by)/c
which are planes,