Problem 777

Solve when $x+y>0$,
$$\Rightarrow\left\{\begin{array}{l} |x|+|y|=2, \\ |x+y|=1 \end{array}\right]\left\{\begin{array}{l} |x|+|1-x|=2, \\ x+y=1 . \end{array}\right.$$

The points $x=0, x=1$ divide the number line into three segments: $(-\infty, 0)$, $[0,1)$, $[1,+\infty)$, yielding the solutions of the system of equations as
$$\left\{\begin{array}{ll} x=\frac{3}{2}, & x=-\frac{1}{2}, \\ y=-\frac{1}{2}, & y=\frac{3}{2} . \end{array}\right.$$

Similarly, when $x+y<0$, the solutions of the system of equations are
$$\left\{\begin{array} { l } { x = - \frac { 1 } { 2 } , } \\ { y = - \frac { 3 } { 2 } , } \end{array} \left\{\begin{array}{l} x=-\frac{3}{2}, \\ y=\frac{1}{2} . \end{array}\right.\right.$$

The above sets of solutions are the solutions to the system of equations.