Say $E$ is a plane cubic, and $p$ is a point on $E$. Riemann-Roch tells us that $\mathcal O_E(3p)$ is very ample.
If $p$ is a flex, it's easy to write down the three sections giving an embedding of $E$ in $\mathbb P^2$: let $L$ be the equation for the tangent line, and use $X/L$, $Y/L$, and $Z/L$. This just gives the embedding of $E$ that you started with.
What happens if $p$ isn't a flex? Is there an easy way to write down the sections of $H^0(E,\mathcal O_E(3p))$? These should give a new embedding, in which $p$ becomes a flex.