My question is how do you solve analitically:
sqrt(x)+y=7 and sqrt(y)+x=11 without solving a 4th degree polynomial equation?
From (1); y<11 and
From (2); x<7 and
Also from (1) and (2) it could be seen that, and
Therefore x=4 is the only solution for x.
Therefore x=4, y=9 is the only solution for this system of equations.
If is rational but not an integer,
Hence is irrational.
Then is irrational. (Contradiction)
Therefore is irrational or an integer.(according to our assumption.)
But if is irrational is irrational. (Contradiction)
So is an integer.
Same proof goes to as well.
Hope this helps.