10.2. Find all values of the parameter $a$, for each of which the system of equations

$$
\left\{\begin{array}{l}
|x|+|y-1|=1 \\
y=a \cdot x+2012
\end{array}\right. \text { has a unique solution. }
$$
Answer. $a= \pm 2011$. Solution. Let's draw on the coordinate plane the set of solutions of the first equation. This is the boundary of a square (see figure). The line representing the second equation passes through the point $M(0 ; 2012)$. Therefore, the condition for the uniqueness of the solution will be satisfied only when the line passes through either point $A(-1 ; 1)$ or point $C(1 ; 1)$. This means that the slope of the line is $\pm 2011$.

![](https://cdn.mathpix.com/cropped/2024_05_06_8f64481553b9b229fb2bg-1.jpg?height=728&width=831&top_left_y=704&top_left_x=1109)