If a, b, and c are non-zero and satisfy a + 1/b = 1 and b + 1/c = 1, determine the exact value of the product abc.

Introduction / Context:
This problem tests manipulation of linked fractional equations and recognition of patterns that eliminate variables. Even though there are two equations with three unknowns, the target expression abc may still be uniquely determined if the relationships are structured appropriately.

Given Data / Assumptions:

• a + 1/b = 1
• b + 1/c = 1
• a, b, c are all non-zero.
• We must compute abc.

Concept / Approach:
Express a and b in terms of c, then multiply. The key is to transform each equation into a rational expression in a single variable. From b + 1/c = 1, we can write b in terms of c; then substitute into a + 1/b = 1 to express a in terms of c. Multiplying a·b·c often causes factors to cancel, leading to a constant independent of c.

Step-by-Step Solution:
From b + 1/c = 1 ⇒ b = 1 − 1/c = (c − 1)/c.Then 1/b = c/(c − 1).From a + 1/b = 1 ⇒ a = 1 − 1/b = 1 − c/(c − 1) = −1/(c − 1).Now compute abc = [−1/(c − 1)] * [(c − 1)/c] * c = −1.

Verification / Alternative check:
Pick a convenient c (e.g., c = 2). Then b = (2 − 1)/2 = 1/2; so a = 1 − 1/b = 1 − 2 = −1. Hence abc = (−1) * (1/2) * 2 = −1, confirming the general result.

Why Other Options Are Wrong:

• 1 and 3 / −3 / −2: These arise from sign mistakes when forming 1/b or from substituting b incorrectly into a.

Common Pitfalls:
Assuming there is not enough information because there are more unknowns than equations; dropping the minus sign when simplifying a = −1/(c − 1).

Final Answer:
-1