10. (20 points) Find all values of the parameter $c$ such that the system of equations has a unique solution

$$
\left\{\begin{array}{l}
2|x+7|+|y-4|=c \\
|x+4|+2|y-7|=c
\end{array}\right.
$$
Answer: $c=3$.

Solution. Let $\left(x_{0} ; y_{0}\right)$ be the unique solution of the system. Then $\left(-y_{0} ;-x_{0}\right)$ also satisfies the conditions of the system. This solution coincides with the first, so $y_{0}=-x_{0}$. The equation $2|x+7|+|x+4|=$ $c$ has a unique solution $x_{0}=-7$ when $c=|7-4|$, since the function $f(x)=2|x+7|+|x+4|$ decreases on the interval $\left(-\infty ; x_{0}\right]$ and increases on $\left[x_{0} ;+\infty\right)$. Similarly, $2|y-7|+|y-4| \geqslant|7-4|$. Therefore, when $c=|7-4|$, the equality $f(x)+f(-y)=2 c$ holds only when $x=-y=-7$, i.e., the system has a unique solution.