jmc

The equation $x^2+y^2-6.5=x+3y$ can be rewritten as $x^2-x+y^2-3y=6.5$. Completing the square and writing decimals as fractions, this can further be rewritten as ${\left(x-\frac{1}{2}\right)}^2-\frac{1}{4}+{\left(y-\frac{3}{2}\right)}^2-\frac{9}{4}=\frac{13}{2}$. Moving the constants to the right side of the equation, this is ${\left(x-\frac{1}{2}\right)}^2+{\left(y-\frac{3}{2}\right)}^2=\frac{10}{4}+\frac{13}{2}=\frac{18}{2}=9$, which is the equation of a circle with center $\left(\frac{1}{2},\frac{3}{2}\right)$ and radius $\boxed{3}$.