# Math Help - solve eq

1. ## solve eq

solve the equations simuntaneousely [1] X +square root of y=11 [2] square root of x+y=7

2. Originally Posted by zaki
solve the equations simuntaneousely [1] X +square root of y=11 [2] square root of x+y=7
Just for completeness here:
$x + \sqrt{y} = 11$
and
$\sqrt{x} + y = 7$

From the bottom equation:
$y = 7 - \sqrt{x}$

Inserting this into the top equation:
$x + \sqrt{7 - \sqrt{x}} = 11$

$\sqrt{7 - \sqrt{x}} = 11 - x$<-- 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.)
$7 - \sqrt{x} = (11 - x)^2$

$\sqrt{x} = x^2 - 22x + 114$

$x = (x^2 - 22x + 114)^2$

$x = x^4 - 44x^3 + 712x^2 - 5016x + 12996$

$x^4 - 44x^3 + 712x^2 - 5017x + 12996 = 0$

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
$y = 7 - \sqrt{x}$

So
$(x, y) = (7.86869, 4.19488), (9, 4), (12.8481, 3.41557), \text{ and } (14.2832, 3.22069)$

-Dan