Solving the System of Equations to Find the Possible Values of abc

In this article, we aim to solve a system of equations to determine the possible values of abc given the conditions:

ab2 9,
bc2 25,
ac2 81
and
abc 1.

Analysis and Solution

A systematic approach will be taken to solve this system by starting with the given equations and analyzing the possible combinations. The steps are as follows:

Step 1: Taking Square Roots

First, we take the square root of each equation. This yields the following results:

ab^2 9 Rightarrow a/b ±3 bc^2 25 Rightarrow b/c ±5 ac^2 81 Rightarrow a/c ±9

Step 2: Analyzing Possible Combinations

We will now analyze various combinations of these results, considering the signs of the equations. Let's start with a few cases:

Case 1: Positive Products

a/b 3 b/c 5 a/c 9

We can express a in terms of b:
a 3b

Substitute a into the equation a/c 9:
3b/c 9 Rightarrow c 6b

Now substitute c into the equation b/c 5:
b(6b) 5 Rightarrow 6b^2 5 Rightarrow 2b^2 5/2 Rightarrow b^2 5/12

Solving for b:
b ±√(5/12)

However, this does not simplify directly to an integer, so let's explore the next valid case:

Case 2: Positive Products and Validation

We continue with the next valid case, where all products are positive:

a/b 3 b/c 5 a/c 9

We substitute a 3b into a/c 9:
3b/c 9 Rightarrow c 3b/9 b/3

Now, substitute c b/3 into b/c 5:
b(b/3) 5 Rightarrow b^2/3 5 Rightarrow b^2 15

Solving for b:
b ±√15

Now, using these values, we can find a and c:

a 3b 3√15
c b/3 √15/3

Finally, calculating the product abc:
abc (3√15)(√15)(√15/3) (3)(15)(1/3) 15

Conclusion

The only valid case, satisfying the condition abc 1:

abc 15
This is the only possible value for abc that satisfies the given conditions.

In summary, the careful analysis of these equations and their sign combinations leads us to conclude that abc 15 is the only valid solution, given the constraints provided.