Answer
$\{\ (3,4),\ (3,-4)\ \}$
Work Step by Step
1. Eliminating:
Eliminate the $(y^{2})$ terms:
$\left\{\begin{array}{lll}
x^{2}+y^{2}=25 & , & /\\
(x-8)^{2}+y^{2}=41 & , & /\times(-1), add
\end{array}\right.$
$x^{2}-(x-8)^{2}=25-41$
2. Solving:
expand the square,
$x^{2}-(x^{2}-16x+64)=-16$
$16x-64=-16\qquad/+64$
$16x=48\qquad/\div 16$
$x=3$
3. Back-substituting into $x^{2}+y^{2}=25$:
$9+y^{2}=25$
$y^{2}=16$
$y=\pm 4$
The solution set is
$\{\ (3,4),\ (3,-4)\ \}$
