Question

The solution space for this system is a subspace of R3 and so must be a line through the origin, a plane through the origin, all of R3, or the origin only. For the given system, determine which is the case. If the subspace is a plane, find an equation for it, and if it is a ine, find parametric equations.

Answer 1

can we see the system?

Answer 2

the system is :

x+5y+10z=0
4x+8y+6z=0
8x+3y-5z=0

Answer 3

The system is homogeneous (constant terms are all zero) which means that there is a solution (compatible system). There are two cases:
a) Rank of coefficients matrix = number of unknowns--->trivial solution x=y=z=0
b) Rank of coefficients matrix < number of unknowns--->indeterminate system(infinite solutions)
The coefficients matrix is as follows:\[\left[\begin{matrix}1 & 5&10 \\ 4& 8&6\\ 8 &3&-5\end{matrix}\right]\] As the determinant is -238 we can say the Rank=3 and the solution is the trivial one, x=y=z=0, that is, the origin