Problem (Algebra):
Let $T=5$. If $|T|-1+3 i=\frac{1}{z}$, compute the sum of the real and imaginary parts of $z$.

Solution:
Let $t=|T|$. Note that $z=\frac{1}{t-1+3 i}=\frac{1}{t-1+3 i} \cdot \frac{t-1-3 i}{t-1-3 i}=\frac{t-1-3 i}{t^{2}-2 t+10}$. Thus the sum of the real and imaginary parts of $z$ is $\frac{t-1}{t^{2}-2 t+10}+\frac{-3}{t^{2}-2 t+10}=\frac{|T|-4}{|T|^{2}-2|T|+10}$. With $T=5$, the answer is $\frac{1}{25}$.

Final answer: $\frac{1}{25}$