Geometry

All the points 5 units away from (2, 7) form a nice little circle. 

The formula for the circle is $(x-2)^2+(x-7)^2=25$. 

We now have a system of equations with

$(x-2)^2+(x-7)^2=25\\ x+y=13$

Let's focus on the first equation. Factoring it, we get

$2x^2-18x+53=25\\ 2x^2-18x+28=0\\ x^2-9x+14=0\\ (x-7)(x-2)=0\\ x=7,2$

Plugging these two values into the second equation, we get 

$y=13-2=11\\ y=13-7=6$

So our two solutions are

$\begin{pmatrix}x=2,\:&y=11\\ x=7,\:&y=6\end{pmatrix}$

So (2,11) and (7,6) are our answers!

Thanks! :)