Cross-multiplying:
5xy=6(x+5), xy=6(y-x), so xy=6(y-x), and 5xy=30y-30x,
5xy=6x+30=30y-30x, so 36x+30=30y, 6x+5=5y, y=(6x+5)/5.
We can now substitute for y in any equation, for example:
xy=6y-6x, x(6x+5)/5=6(6x+5)/5-6x, multiply through by 5:
x(6x+5)=6(6x+5)-30x=36x+30-30x=6x+30,
6x2+5x=6x+30, 6x2-x-30=0, x=(1±√(1+720))/12 (quadratic formula).
So x=(1+√721)/12 or (1-√721)/12.
y=(6x+5)/5=(11+√721)/10 or (11-√721)/10.
SOLUTION (x,y)=((1+√721)/12,(11+√721)/10)) or ((1-√721)/12,(11-√721)/10)).
This can be written (x,y)=(2.3210,3.7851) or (-2.1543,-1.5851) approx.
The graph below shows xy/(x+5)=6/5 (red), xy/(y-x)=6. The two curves intersect at the points shown (green and blue intersections).