Let's define variables for each drink:

• $b$: cost of one beer
• $w$: cost of one glass of wine
• $c$: cost of one cappuccino
• $h$: cost of one whiskey

Given relationships:

1. The price of a beer is three times the price of a cappuccino: $b = 3c$
2. The price of a whiskey is three times the price of a wine: $h = 3w$

First round of drinks:

• 3 beers, 2 glasses of wine, 7 cappuccinos, and 3 whiskeys cost $103: $3b + 2w + 7c + 3h = 103$

Second round of drinks:

• 9 beers, 1 glass of wine, 4 cappuccinos, and 5 whiskeys cost $173: $9b + 1w + 4c + 5h = 173$

Substituting the relationships for $b$ and $h$ into the two equations:

1. Substituting $b = 3c$ and $h = 3w$ into the first equation: $3(3c) + 2w + 7c + 3(3w) = 103$ Simplifying: $9c + 2w + 7c + 9w = 103$ $16c + 11w = 103$

2. Substituting $b = 3c$ and $h = 3w$ into the second equation: $9(3c) + 1w + 4c + 5(3w) = 173$ Simplifying: $27c + 1w + 4c + 15w = 173$ $31c + 16w = 173$

System of linear equations:

We now have the system:

1. $16c + 11w = 103$
2. $31c + 16w = 173$

Using Cramer's rule, we solve for $w$ (the cost of one glass of wine).

The general formula for Cramer's rule to solve for $w$ is: $w = \frac{\text{determinant of the matrix with the w column replaced by the constants}}{\text{determinant of the coefficient matrix}}$

Let's compute it.The cost of one glass of wine is $5$.

Would you like further details or have any questions? Here are five related questions to expand on this:

1. How is Cramer's rule applied to larger systems of equations?
2. What would be the price of a cappuccino based on the solution?
3. Can Cramer's rule be used for non-square systems of equations?
4. What is the significance of the determinant in linear algebra?
5. How does the relationship between the drinks’ prices affect the system of equations?

Tip: Cramer's rule works efficiently for small systems but becomes computationally expensive for larger ones, where other methods like Gaussian elimination are preferred.