I4.1 Given that $x$ and $y$ are real numbers such that $|x|+x+y=10$ and $|y|+x-y=10$. If $P=x+y$, find the value of $P$.
If $x \geq 0, y \geq 0 ;\left\{\begin{aligned} 2 x+y & =10 \\ x & =10\end{aligned}\right.$, solving give $x=10, y=-10$ (reject) If $x \geq 0, y<0,\left\{\begin{array}{l}2 x+y=10 \\ x-2 y=10\end{array}\right.$, solving give $x=6, y=-2$
If $x<0, y \geq 0,\left\{\begin{array}{l}y=10 \\ x=10\end{array}\right.$, reject
If $x<0, y<0 ;\left\{\begin{aligned} y & =10 \\ x-2 y & =10\end{aligned}\right.$, solving give $x=30, y=10$ (reject)
$$
\begin{array}{l}
\therefore x=6, y=-2 \\
P=x+y=6-2=4
\end{array}
$$