All lines of the family $5x+3y-2=0$ and $\left(3x-y-4\right)=0$ are concurrent.

The point of concurrency is $\left(1,-1\right)$
Similarly, the point of concurrency of lines of  the other family is $\left(\mathrm{3,4}\right)$
$\Rightarrow$ The line of both families is the line passing through $\left(1,-1\right) \& \left(\mathrm{3,4}\right)$
$\Rightarrow$ Equation of the required line is $5x-2y-7=0$