13.132. In a laboratory setup, a certain liquid flows into a vessel through three inlet valves. If all valves are opened simultaneously, the vessel will be filled in 6 minutes. If the vessel is filled only through the second valve, it will take 0.75 of the time it takes to fill the vessel only through the first valve. Filling the vessel through the third valve alone takes 10 minutes longer than through the second valve. For how long does each valve need to be opened individually to fill the vessel?
## Solution.

Let the first tap be opened for $x$ min, the second for $y$ min, and the third for $z$ min. If all taps are opened simultaneously, then $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{6}$. According to the conditions, $y=0.75 x ; z=y+10$. Solving the system

$$
\left\{\begin{array}{l}
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{6} \\
y=0.75 x, \\
z=y+10,
\end{array} \text { we find } x=\frac{56}{3} \text { min; } y=14 \text { min; } z=24\right. \text { min. }
$$

Answer: for $\frac{56}{3}, 14$ and 24 min.