Show that $\textsf W$ is generated by $u$ and $v$

Question

Let $\textsf W=\{(a,b,0): \, a,b\in \Bbb R \}$ in $\mathbb{R}^{3}$ and let $u=(2,-1,0)$ and $v=(1,3,0)$. Show that $\textsf W$ is generated by $u$ and $v$. I attempted to show this by $$(a,b,0)=x(2,-1,0)+y(1,3,0)$$ and I got $$2x+y=a$$ $$-x+3y=b$$ $$0=0$$ by manipulating the equations I got $$y=\frac{2a+b}{7} \notin \mathbb{R}$$ Where did I go wrong?

Answer 1

$y=\frac {a+2b} 7$ and $x=\frac {3a-b} 7$. Both of these are of course real numbers.