Find the sum of all values of $x$ for which the expression $\frac{x-3}{x^2-10x+16}$ is undefined.
The given expression is undefined when the denominator is equal to zero. Thus, we need to find the solutions to the equation $x^2-10x+16=0$. Factoring, we see that $(x-8)(x-2)=0$, so the solutions are $x=8$ and $x=2$. The sum of these solutions is $\boxed{10}$.
To determine when a rational expression is undefined, we need to find the values of $x$ that make the denominator equal to zero. In this case, we have the denominator $x^2 - 10x + 16$. To solve $x^2 - 10x + 16 = 0$, we can either factor the quadratic or use the quadratic formula.
Factoring the quadratic gives us $(x-2)(x-8) = 0$. Setting each factor equal to zero gives us two possible values for $x$ that would make the denominator zero: $x-2 = 0$ or $x-8 = 0$, which gives $x = 2$ and $x = 8$.
Therefore, the values of $x$ for which the expression $\frac{x-3}{x^2-10x+16}$ is undefined are $x = 2$ and $x = 8$. To find the sum of these values, we simply add them together: $2 + 8 = 10$.
Hence, the sum of all values of $x$ for which the expression is undefined is $10$.