1. Solve the system of equations

$$
\left\{\begin{array}{l}
|x|+|y-1|=1 \\
x+2 y=3
\end{array}\right.
$$
Solution. a) If $x \geq 0, y \geq 1$ we get the system

$$
\left\{\begin{array}{l}
x+y-1=1 \\
x+2 y=3
\end{array}\right.
$$

whose solution is $(1,1)$. If $x \geq 0, y<1$, we get the system

$$
\left\{\begin{array}{l}
x-y+1=1 \\
x+2 y=3
\end{array}\right.
$$

and this system has no solution in this region. If $x<0, y \geq 1$ we get the system

$$
\left\{\begin{array}{c}
-x+y-1=1 \\
x+2 y=3
\end{array}\right.
$$

whose solution is $\left(-\frac{1}{3}, \frac{5}{3}\right)$. If $x<0, y<1$ we get the system

$$
\left\{\begin{array}{c}
-x-y+1=1 \\
x+2 y=3
\end{array}\right.
$$

and this system has no solution in this region.