Let $abcdef$ denote any such six-digit number and convert each requirement in the problem statement into an equation.
\begin{align*}
a&=b-1\\
c&=\frac{1}{2}b\\
d&=3c\\
10e+f&=d+e\\
24&=a+b+c+d+e+f
\end{align*}
In a more standard form this becomes
\begin{align*}
a-b&=-1\\
-b+2c&=0\\
-3c+d&=0\\
-d+9e+f&=0\\
a+b+c+d+e+f&=24
\end{align*}
Using equation operations (or the techniques of the upcoming Section RREF), this system can be converted to the equivalent system
\begin{align*}
a+\frac{16}{75}f &= 5\\
b+\frac{16}{75}f &= 6\\
c+ \frac{8}{75}f &= 3\\
d+ \frac{8}{25}f &= 9\\
e+\frac{11}{75}f &= 1
\end{align*}
Clearly, choosing $f=0$ will yield the solution $abcde=563910$. Furthermore, to have the variables result in single-digit numbers, none of the other choices for $f$ ($1,\,2,\,\ldots,\,9$) will yield a solution.