Find the geometric interpretation of general solution 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! 

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?
