The positive real numbers $a,b,c$ satisfy:

What is the value of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$?

Step 1: Let the two given equations be:

Step 2: Rearrange the terms of equation (2) to group them with the corresponding denominators in (1):

Step 3: Add equations (1) and (2) together:

Step 4: Since $a,b,c$ are positive real numbers, all terms in the sum are positive. By the Arithmetic Mean-Geometric Mean (AM-GM) inequality:

Step 5: Since the sum is exactly $3$, the equality condition of the AM-GM inequality must hold. This requires all three terms to be equal to $1$:

This implies:

for some positive constant $k$.

Step 6: Substitute these into equation (2):

Step 7: Now solve for $a,b,c$:

Step 8: Finally, calculate the requested sum: