Basis from 2 equations

Question

$V$ is the subspace in $\mathbb{R}^3$ consisting of vectors $(x,y,z)$ satisfying $x+2y-3z=0$ and $y-2z=0$. How do I find the basis of this space? I tried using RREF but it doesn't make sense. Is it the row space I should be looking at? That makes $(1,2,-3)$ and $(0,1,-2)$ the basis. How do I think about this?

Answer 1

Intersection of two planes is a line. All you have to do is find one non -zero vector in the line. One such vector is $(-1,2,1)$. So $\{(-1,2,1)\}$ is a basis.