## Task 1 - 331221

Determine all pairs $(x ; y)$ of real numbers that satisfy the following system of equations (1), (2)!

$$
\begin{array}{r}
2-x+y=\sqrt{18+x-y} \\
\sqrt{1+x+y}+\sqrt{2+x-y}=5
\end{array}
$$
By squaring (1), one first obtains

$$
(x-y)^{2}-4(x-y)+4=18+x-y
$$

and through a simple transformation, this further becomes

$$
(x-y)^{2}-5(x-y)-14=(x-y+2)(x-y-7)=0
$$

Of the two possibilities

$$
x-y=-2 \quad \text{or} \quad x-y=7
$$

only

$$
x-y=-2
$$

is indeed a solution of (1), as one can immediately verify by substitution. The second is instead a false solution introduced by the squaring, as it would lead to the contradiction $-5=\sqrt{25}$. By substituting $x-y=-2$, (2) simplifies immediately to

$$
\sqrt{1+x+y}=5 \quad \text{or} \quad x+y=24
$$

from which, in conjunction with (3), one immediately obtains

$$
x=11, y=13
$$

which indeed solves the system of equations (1) and (2) here.