x4y5+x5y4=810

x3y6+x6y3=945

{x,y}=?

We have, x4y4(x+y)=810 and
x3y3(x3+y3)=945 or x3y3(x+y)(x2xy+y2)=945

On division, x2xy+y2xy=945810=76

xy1+yx=76

Let, yx=v

So, 1v1+v=76

v+1v=76+1=136

Thus, 6v2+6=13v

6v24v9v+6=0

(3v2)(2v3)=0

v=32,23

Since, v=yx=32, y=3k & x=2k

So, (2k)4.(3k)4.(2k+3k)=810

8k9=1

k=121/3

Now, y=3k=3.21/3 and x=2k=2.21/3=22/3

When v=23, it can be shown that the values of x and y would interchange, that is x=3.21/3 and y=22/3.

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