Prime System

$\Delta = \begin{vmatrix} 2 &3 \\ 7 &11 \end{vmatrix} = 1$

$\Delta_{\text{a}} = \begin{vmatrix} 5 &3 \\ 13 &11 \end{vmatrix} \Rightarrow \text{a} = 16$

$\Delta_{\text{b}} = \begin{vmatrix} 2 &5 \\ 7 &13 \end{vmatrix} \Rightarrow \text{b} = -9$

$17\text{a} + 19\text{b} = 17 \cdot 16 + 19 \cdot (-9)$ $\Rightarrow \boxed{ 17\text{a} + 19\text{b}= 101.}$

do we really need to write its solution.. its almost obvious

Just a simple little system. Multiply equation one by 3.5, subtract one from two, find that b is negative 9, plug in and find a is 16, then plug into 17a + 19b to get 101.

Since $2a + 3b = 5$ and $7a + 11b = 13$ , $14a + 21b = 35$ and $14a + 22b = 26$ so $b$ equals $-9$ . By inputting the value of $b$ you can find that the value of $a$ is 16. $(17 \times 16) + (19 \times -9)$ equals to $101$ .