jmc

Multiplying both sides by $x$ and then subtracting $10x$ from each side gives $x^2-10x+25=0.$ The quadratic factors to give $(x-5{)}^2=0,$ so $x-5=0,$ and $x=5$ is the only solution. Thus the answer is $\boxed{5}.$

Note: We might want to use the fact that the sum of the solutions to a quadratic $a{x}^2+bx+c=0$ is given by $-b/a,$ but be careful! That fact counts double roots twice for the purpose of the sum, but this problem only counts it once, since $x=5$ is the only solution.