Solve the following system of equations:

$$
\begin{gathered}
x + xy = 19 - y \\
\frac{84}{xy} - y = x
\end{gathered}
$$
$$
x + xy = 19 - y
$$

$$
\frac{84}{xy} - y = x
$$

By adding the two equations, we get:

$$
\frac{84}{xy} + xy - 19 = 0
$$

$$
(xy)^2 - 19xy + 84 = 0
$$

From which

$$
xy = 12, x'y' = 7
$$

Substituting these values of \(xy\) into (1), we get:

$$
x + y = 7, x' + y' = 12
$$

Solving the equations under (4) and (5), we get:

$$
\begin{aligned}
& x_1 = 3, x_2 = 4, x_3 = 6 + \sqrt{29}, x_4 = 6 - \sqrt{29} \\
& y_1 = 4, y_2 = 3, y_3 = 6 - \sqrt{29}, y_4 = 6 + \sqrt{29}
\end{aligned}
$$

(Béla Kiss, student at Ludovika Academy).

The problem was also solved by: László Fleischmann, state rabbinical training institute V. o. t., Budapest; Bernát Friedmann, main gymnasium VII. o. t., S.-A.-Ujhely; Főr. VII. o. t., Pécs; Ervin Hofbauer, Kántor Nándor and Pál István Langheim, main gymnasium VII. o. t., Budapest; Gyula Pósch, main gymnasium VIII. o. t., Budapest; Jenő Schiller, main gymnasium VIII. o. t., Győr; Józef Suschnik, main gymnasium VIII. o. t., Kecskemét; Gusztáv Szabó, main gymnasium VIII. o. t., Győr; Aladár Visnya, main gymnasium VIII. o. t., Pécs; Győző Zemplén, main gymnasium VIII. o. t., Fiume.