We are given that $$\sqrt{2x}+\sqrt{y}=13,$$$$\sqrt{8x}+\sqrt{9y}=35.$$ We can rewrite this as $$\sqrt{2} \cdot \sqrt{x} + \sqrt{y}=13,$$$$2\sqrt{2} \cdot \sqrt{x} + 3 \cdot \sqrt{y} = 35.$$Let $$a=\sqrt{x}, b=\sqrt{y}.$$We have $$\sqrt{2}a + b = 39,$$$$2\sqrt{2}a + 3b = 35.$$Multiplying the first equation by $-3$ yields $$-3\sqrt{2}a-3b=-39.$$Adding this to the second equation gives $$-\sqrt{2}a=-4 \rightarrow a = 2\sqrt{2} \rightarrow b = 9 \rightarrow (x,y)=(8,81) \rightarrow 20x+23y=160+1863=\boxed{2023}.$$