If the system of equations

$\mathrm{x}+\mathrm{y}+\mathrm{a} \mathrm{z}=\mathrm{b}$

$2 x+5 y+2 z=6$

$\mathrm{x}+2\mathrm{y}+3\mathrm{z}=3$

has infinitely many solutions, then 2a + 3b is equal to

Given, $\mathrm{x}+\mathrm{y}+\mathrm{az}=\mathrm{b}$

$2\mathrm{x}+5\mathrm{y}+2\mathrm{z}=6\mathrm{x}+2\mathrm{y}+3\mathrm{z}=3$

$\mathrm{\Delta }=0\Rightarrow \left|\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}a\\ 2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}5\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}3\end{array}\right|=0$

$$\begin{gathered} 1(15-4)-1(6-2)+\mathrm{a}(4-5)=0 \\ \Rightarrow 11-4-\mathrm{a}=0 \\ \Rightarrow \mathrm{a}=7 \end{gathered}$$

$$\begin{gathered} {\mathrm{\Delta }}_1=0\Rightarrow \left|\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{b}\\ 2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}5\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}6\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}3\end{array}\right|=0 \\ \Rightarrow 1(15-12)-1(6-6)+\mathrm{b}(4-5)=0 \end{gathered}$$

$b=3$

$2\mathrm{a}+3\mathrm{b}=2\left(7\right)+3\left(3\right)=14+9=23$