2. Determine all pairs $(a, b)$ of real parameters for which the system of equations

$$
\begin{aligned}
|x|+y & =a \\
2|y|-x & =b
\end{aligned}
$$

has exactly three solutions in the set of real numbers, and for each of them, determine these solutions.
SOLUTION. Instead of the usual algebraic method, we will prefer the graphical method of solving the given system of equations. For this purpose, we rewrite it in the form $y=-|x|+a, x=2|y|-$ - $b$. From the graphs of both these dependencies between the variables $x$ and $y$ (for fixed numbers $a$ and $b$), examples of which (for chosen $a, b$) are drawn ${ }^{1}$ in Fig. 1, we can see (considering the angles that the tangent half-lines form with the coordinate axes) that in the case where both numbers $a, b$ are non-positive, the given system of equations has no solutions except for the case $a=b=0$, when the solution is unique $(x=y=0)$.

![](https://cdn.mathpix.com/cropped/2024_04_17_bc2a4ef504c610875539g-2.jpg?height=451&width=763&top_left_y=951&top_left_x=652)

Fig. 1

Furthermore, we see that in the case $a>0, b \leqq 0$ the system has at most two solutions, just as in the case $a \leqq 0, b>0 .^{2}$ Finally, in the remaining case $a>0, b>0$ it is not difficult to conclude that the system will have exactly three solutions only if the graphs of both dependencies are in one of the relative positions shown in Figures 2 and 3. They express

![](https://cdn.mathpix.com/cropped/2024_04_17_bc2a4ef504c610875539g-2.jpg?height=596&width=757&top_left_y=1735&top_left_x=227)

Fig. 2

![](https://cdn.mathpix.com/cropped/2024_04_17_bc2a4ef504c610875539g-2.jpg?height=591&width=734&top_left_y=1738&top_left_x=1095)

Fig. 3

1 If we change the values of the parameters $a$ and $b$, both graphs will move in the direction of the respective coordinate axis. The idea of these two (mutually independent) movements entitles us to conclusions about the possible relative positions of both graphs, as further stated in this solution without further explanation.

2 For each of the two cases, draw situations where the system has 0, 1, or 2 solutions.
i corresponding numbers $a$ and $b$ using a positive parameter $k$. The three common points of both graphs are marked with empty circles on both figures; their coordinates $(x, y)$, which we can easily calculate, are then the sought-after triple solutions of the given system.

Conclusion. The given system of equations has exactly three solutions for every pair of parameters $(a, b)=(k, 2 k)$ and for every pair of parameters $(a, b)=(k, k)$, where $k$ is any positive real number. In the first case, the given system has solutions $(0, k),\left(-\frac{4}{3} k,-\frac{1}{3} k\right)$ and $(4 k,-3 k)$. In the second case, the given system has solutions $(-k, 0),\left(\frac{1}{3} k, \frac{2}{3} k\right)$ and $(3 k,-2 k)$.