Answer: $19$

Elimination multiplying equations by $3$ and $8$

               $\begin{align}3g+8b=46\quad\Rightarrow\quad&24g+64b=368\\

8g+3b=31\quad\Rightarrow\quad&\ \ \underline{24g+9b=\ \ 93}\\

\text{subtracting gives:}\quad&\ \ \ \qquad 55b=275\\

&\ \ \ \Rightarrow\ \ 11b=55\\

& \ \ \ \Rightarrow\quad\ \ b=5\end{align}$

And so $3g+40=46\Rightarrow g=2$

$\therefore 2g+3b= 4+15=19$

Elimination by combining equations more than once

$$\begin{align} 3g+8b=46&\\

\underline{+\qquad 8g+3b=31}&\\

11g+11b=77&\\

\Rightarrow \quad g+b=\ \ 7&\\

\Rightarrow 3g+3b=21\end{align}$$

                     $\begin{align} 3g+8b=46&\\

\underline{-\qquad 3g+3b=21}&\\

5b=25&\\

\Rightarrow \quad b=\ 5\ &\end{align}$              $\begin{align} 8g+83=31&\\

\underline{-\qquad 3g+3b=21}&\\

5g=10&\\

\Rightarrow \quad g=\ 2\ &\end{align}$

Therefore $2g + 3b = 19$.

Getting $2g+3b$ without finding $g$ and $b$

$\begin{align} 3g+8b=46&\\

\underline{+\qquad 8g+3b=31}&\\

11g+11b=77&\\

\Rightarrow \quad g+b=\ \ 7& \end{align}$

        $3g+8b$

$+ \quad (\ g\ +\ b\ )\times$ some    to get blues and greens in ratio $2:3$

                                    difference will be $5$ since $8$ and $3$ have difference of $5$

                                    $10:15 = 3+7:8+7$

$\begin{align} 3g+8b=46&\\

\underline{+\qquad 7g+7b=49}&\\

10g+15b=95&\\

\Rightarrow \ \ 2g+3b=19& \end{align}$