B2. Find all pairs of real numbers $x$ and $y$ that solve the system of equations

$$
\begin{aligned}
\frac{3}{x-4 y}+\frac{2}{x+y-5} & =0 \\
\frac{2}{x^{2}-4 y^{2}}+\frac{1}{x^{2}+y^{2}-5} & =0
\end{aligned}
$$

## 61st Mathematical Competition for High School Students in Slovenia

Selection Competition, March 16, 2017

## Problems for 2nd Year

## Time for solving: 45 minutes.
B2. Eliminate the fractions to get

$$
\begin{aligned}
3(x+y-5)+2(x-4 y) & =0 \\
2\left(x^{2}+y^{2}-5\right)+\left(x^{2}-4 y^{2}\right) & =0
\end{aligned}
$$

and simplify both equations to

$$
\begin{array}{r}
5 x-5 y-15=0 \\
3 x^{2}-2 y^{2}-10=0
\end{array}
$$

From the first equation, express $x=y+3$. When we substitute this into the second equation, we get $y^{2}+18 y+17=0$. Factor the left side of the equation to get $(y+1)(y+17)=0$. The solutions are $y=-1$ and $y=-17$. In the first case, from the relation $x=y+3$ we get $x=2$, and in the second case $x=-14$. We test both solutions and find that in the case $x=2$, $y=-1$, the fractions $\frac{2}{x^{2}-4 y^{2}}$ and $\frac{1}{x^{2}+y^{2}-5}$ are not defined. The only solution to the system is thus the pair $x=-14, y=-17$.

The system of equations without fractions .....................................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_9b375d7a0e2f6834f525g-12.jpg?height=55&width=1573&top_left_y=1126&top_left_x=276)

![](https://cdn.mathpix.com/cropped/2024_06_07_9b375d7a0e2f6834f525g-12.jpg?height=60&width=1579&top_left_y=1169&top_left_x=273)

![](https://cdn.mathpix.com/cropped/2024_06_07_9b375d7a0e2f6834f525g-12.jpg?height=55&width=1579&top_left_y=1229&top_left_x=273)

![](https://cdn.mathpix.com/cropped/2024_06_07_9b375d7a0e2f6834f525g-12.jpg?height=66&width=1579&top_left_y=1275&top_left_x=273)

![](https://cdn.mathpix.com/cropped/2024_06_07_9b375d7a0e2f6834f525g-12.jpg?height=63&width=1579&top_left_y=1325&top_left_x=273)

![](https://cdn.mathpix.com/cropped/2024_06_07_9b375d7a0e2f6834f525g-12.jpg?height=60&width=1579&top_left_y=1375&top_left_x=273)

![](https://cdn.mathpix.com/cropped/2024_06_07_9b375d7a0e2f6834f525g-12.jpg?height=60&width=1579&top_left_y=1421&top_left_x=273)

![](https://cdn.mathpix.com/cropped/2024_06_07_9b375d7a0e2f6834f525g-12.jpg?height=54&width=1579&top_left_y=1481&top_left_x=273)

## 61st Mathematical Competition for High School Students in Slovenia

Selection Competition, March 16, 2017

Each mathematically correct and complete solution is worth 20 points, even if the solving process is different from the official solution.

Solutions for 2nd year students