Find the geometric interpretation of general solution

Question

Here are the linear equation: $$2x+z=0$$ $$-x+3y+z=0$$ $$-x+y+z=0$$

I have found that the general solution is, $$t \begin{bmatrix} \frac{-1}2\\ \frac{-1}2 \\ 1 \\ \end{bmatrix} $$

The question asks me to find the geometric interpretation of general solution. But I have no idea how. I think my limit knowledge of geometric interpretation is not helping me, so some explanation about geometric interpretation would help me a lot. Thanks you for any help!

Answer 1

What you have is the parametric equation for a line. Do you see why?

Answer 2

Your solution:

$$t \begin{bmatrix} \frac{-1}2\\ \frac{-1}2 \\ 1 \\ \end{bmatrix} $$

represents every multiple of the single vector $[-1/2,-1/2,1]^T$. That one vector represents a point in $\mathbb{R}^3$. Think of it, and all of its multiples, as a set of points in $\mathbb{R}^3$. Positive multiples (positive values of $t$) are on the same side of the origin as the original vector; negative multiples are on the opposite side of the origin. The zero-multiple (taking $t=0$) is the origin.

Do you see how this is a description of a line through the origin?