$2^{x}+3^{y}=68\times 2^{x}-3^{y+1}\text{ and }68\times 2^{x}-3^{y+1}=5$

$x=\frac{\ln(3)y+\ln(\frac{4}{67})}{\ln(2)}$
$\frac{\ln(3)y-\ln(67)}{\ln(2)}+2=\log_{2}\left(\frac{3\times 3^{y}+5}{17}\right)-2$

$y=\frac{\ln(2)x+\ln(\frac{67}{4})}{\ln(3)}$
$x>\frac{\ln(\frac{5}{68})}{\ln(2)}\text{ and }\frac{\ln(2)x+\ln(\frac{67}{4})}{\ln(3)}=\log_{3}\left(68\times 2^{x}-5\right)-1$