Question

$\sqrt{x} +y=7 and x+ \sqrt{y} =11.Find the values of x and y.$

Answer 1

From equation 1:
$y=7- \sqrt{x}$
Put this in the second equation:
$x+ \sqrt{7- \sqrt{x} } = 11$
$\sqrt{7- \sqrt{x} } = 11-x \\ 7- \sqrt{x} =121+ x^{2} -22x \\ (x^2-22x+114)^2=x$

If the above equation is solved, it will give x = 
 9.0000
14.2832
 12.8481
  7.8687

and then y can be found out very easily

Answer 2

$\sqrt{x} + y=7$
$y=7- \sqrt{x}$
Putting y as 7-√x from eqn.(1) into eqn (2) we get:
$x+ \sqrt{7- \sqrt{x} }$=11
⇒$(x+ \sqrt{7- \sqrt{x})^{2}$=121
Let 7-√x be b.
⇒$(x+ \sqrt{b} )^{2} =121$
⇒$x^{2} +2 \sqrt{b}x$+b=121.
Solving this equation gives x=9 and 
81+18√b+b=121
⇒√b(18+√b)=40
Now solve and find out √b.