r = red one.
g = green one.
b = blue one.
[tex]\bf \begin{cases}
r+g=24\\
r+b=10\\
b+g=20
\end{cases}\\\\
-------------------------------\\\\
r+g=24\implies \boxed{r}=24-g
\\\\\\
r+b=10\implies \boxed{24-g}+b=10\implies b=10-24+g
\\\\\\
\underline{b}=-14+g\\\\
-------------------------------[/tex]
[tex]\bf therefore\qquad b+g=20\implies \underline{-14+g}+g=20
\\\\\\-14+2g=20
\\\\\\
2g=34\implies g=\cfrac{34}{2}\implies g=17\qquad
\begin{cases}
b+g=20\\
b=20-g\\
b=20-17\\
b=3\\
------\\
r+b=10\\
r+3=10\\
r=10-3\\
r=7
\end{cases}
\\\\\\
\stackrel{7}{r}+\stackrel{b}{3}+\stackrel{17}{g}=27[/tex]
