1. The problem states that Flannery used 30 lilies and 78 roses to create six identical flower arrangements. 2. We need to write an equation relating $l$, the number of lilies, and $r$, the number of roses, in one arrangement. 3. Since the total lilies and roses are divided equally into 6 arrangements, each arrangement has $\frac{30}{6}$ lilies and $\frac{78}{6}$ roses. 4. Simplify these fractions: $\frac{30}{6} = 5$ and $\frac{78}{6} = 13$. 5. Therefore, each arrangement has 5 lilies and 13 roses. 6. The relationship between $l$ and $r$ for one arrangement is $l = 5$ and $r = 13$. 7. To express the relationship as an equation, since the ratio of lilies to roses is constant, we write $\frac{l}{5} = \frac{r}{13}$. 8. This equation shows that the number of lilies and roses in any arrangement maintain the ratio 5 to 13. Final equation: $$\frac{l}{5} = \frac{r}{13}$$