Biggest x+y ever

$3x=\dfrac7y$ $3xy=7$ $3y=\dfrac7x$ Substitute the equation above into $6x-19=3y$ $6x-19=\dfrac7x$ $x\left(6x-19\right)=7$ $6x^2-19x=7$ $6x^2-19x-7=0$ $\left(2x-7\right)\left(x+1\right)=0$ We need the biggest value of x and y.Thus, $x=\dfrac72$ When $x=\dfrac72$ , $3\left(\dfrac72\right)=\dfrac7y$ $\dfrac{21}{2}=\dfrac7y$ $\dfrac{2}{21}=\dfrac{y}{7}$ $y=7\times\dfrac{2}{21}$ $y=\dfrac23$ $x+y=\dfrac72+\dfrac23$ $=\dfrac{25}{6}$ $=4.1666...$ $\approx\boxed{4.167}$