3d6c9f…: “Toward a gray tetrahedron”

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The linear system associated to the augmented matrix below is consistent.
$\begin {amatrix}{3} a&b&c &0\\ d&e&f&0\\ g &h&i&0 \end {amatrix}$

True False

Given the RREF augmented matrix below, write the solution to the associated system in parametric vector form.
$\begin {amatrix}{3} 1&0&9&-8\\ 0&1&-4&5 \end {amatrix}$

Answer: $\bx = \left [ \begin {array}{c} \answer {-8} \\ \answer {5} \\ \answer {0} \end {array} \right ] +x_3 \left [ \begin {array}{c} \answer {-9}\\ \answer {4}\\ \answer {1} \end {array} \right ]$

A system of linear equations is called homogeneous if it can be written in the form $\dots$

$A\bx = {\bf 0}$ Ax=b

You are given a homogeneous system with no free variables. How many solutions does it have?

none one infinitely many not enough information to know

Geometrically, the parametric vector form of a solution $\bx = \bp + t\bv$ represents:

a line through the origin parallel to $\bp$ . a line through the origin parallel to $\bv$ . a line through $\bv$ parallel to $\bp$ . a line through $\bp$ parallel to $\bv$ . a plane through $\bp$ and $\bv$ .