Given that

$$
\begin{aligned}
a+b & =23 \\
b+c & =25 \\
c+a & =30
\end{aligned}
$$

determine (with proof) the value of $a b c$.
We add all three equations to obtain

$$
2(a+b+c)=78
$$

so $a+b+c=39$. Therefore,

$$
\begin{aligned}
& a=39-25=14 \\
& b=39-30=9 \\
& c=39-23=16 .
\end{aligned}
$$

Thus $a b c=14 \cdot 9 \cdot 16=2016$.