6.009. $\frac{5 a}{y+a}+\frac{4 a}{y+2 a}+\frac{3 a}{y+3 a}=8$.
Solution.

Domain of definition: $y \neq -a, y \neq -2a, y \neq -3a$.

$$
\frac{5a}{y+a} + \frac{4a}{y+2a} + \frac{3a}{y+3a} = 8 \Leftrightarrow \frac{8y^3 + 36ay^2 + 38a^2y}{(y+a)(y+2a)(y+3a)} = 0
$$

or

$$
\begin{aligned}
& \left\{\begin{array}{l}
y(4y^2 + 18ay + 19a^2) = 0, \\
y \neq -a, \\
y \neq -2a, \\
y \neq -3a
\end{array}\right. \\
& \Rightarrow y = 0 \text{ or } 4y^2 + 18ay + 19a^2 = 0.
\end{aligned}
$$

From this, $y_{1,2} = \frac{-9a \pm \sqrt{81a^2 - 76a^2}}{4} = \frac{-9a \pm a\sqrt{5}}{4} = \frac{a(-9 \pm \sqrt{5})}{4}$.

Answer: $y_1 = 0, y_{2,3} = \frac{a(-9 \pm \sqrt{5})}{4}$.