solve the equations simuntaneousely [1] X +square root of y=11 [2] square root of x+y=7
Just for completeness here:
$\displaystyle x + \sqrt{y} = 11$
and
$\displaystyle \sqrt{x} + y = 7$
From the bottom equation:
$\displaystyle y = 7 - \sqrt{x}$
Inserting this into the top equation:
$\displaystyle x + \sqrt{7 - \sqrt{x}} = 11$
$\displaystyle \sqrt{7 - \sqrt{x}} = 11 - x$<-- Square both sides
Since the solution is going to turn out to be ugly anyway, I'm going to use a trick. (They use it in the referred thread as well.)
$\displaystyle 7 - \sqrt{x} = (11 - x)^2$
$\displaystyle \sqrt{x} = x^2 - 22x + 114$
$\displaystyle x = (x^2 - 22x + 114)^2$
$\displaystyle x = x^4 - 44x^3 + 712x^2 - 5016x + 12996$
$\displaystyle x^4 - 44x^3 + 712x^2 - 5017x + 12996 = 0$
Using the rational root theorem, you can find one nice solution: x = 9. You can get the rest numerically: x = 7.86869, 12.8481, 14.2832.
Thus
$\displaystyle y = 7 - \sqrt{x}$
So
$\displaystyle (x, y) = (7.86869, 4.19488), (9, 4), (12.8481, 3.41557), \text{ and } (14.2832, 3.22069)$
-Dan