If we have $ax+by+cz = 0$ show that $span(x,y)=span(y,z)$
My steps:
$$x = -\frac{b}{a}y-\frac{c}{a}z\Rightarrow x\ \in\ span(y,z)$$
$$z=-\frac{a}{c}x-\frac{b}{c}y\Rightarrow z \in\ span(x,y)$$
But we still have not proved that $span(x,y)=span(y,z)$ and I'm not entirely sure how to proceed from here
I'll assume $abc\neq 0$.
As you pointed out, $x\in \text{span}(y,z)$, $z\in \text{span}(x,y)$, and $y\in \text{span}(x,z)$. This means $$ \text{span}(y,z)\subseteq \text{span}(y,\text{span}(x,y))=\text{span}(x,y) $$and likewise $$ \text{span}(x,y)\subseteq \text{span}(y,\text{span}(y,z))=\text{span}(y,z) $$Another way: the equation is a $2$-dimensional plane, so it should be spanned by any combination of two of the basis vectors.