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## Main Question or Discussion Point

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

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

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A plane

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HallsofIvy

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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,

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,

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