To solve the problem using Cramer's rule, we'll follow these steps:

Step 1: Define the variables

Let's denote the prices as follows:

• Let $b$ be the price of one beer.
• Let $w$ be the price of one glass of wine.
• Let $c$ be the price of one cappuccino.
• Let $h$ be the price of one whiskey.

Step 2: Set up the equations

From the problem statement, we have the following information:

1. $b = 2c$ (The price of a beer is double the price of a cappuccino)
2. $h = 2w$ (The price of whiskey is double the price of wine)

We also have the cost equations: $4b + 2w + 7c + 3h = 85 \quad \text{(Equation 1)}$ $9b + 1w + 4c + 5h = 121 \quad \text{(Equation 2)}$

Step 3: Substitute the relationships into the equations

Using $b = 2c$ and $h = 2w$, we substitute into the two equations:

Equation 1: $4(2c) + 2w + 7c + 3(2w) = 85$ $8c + 2w + 7c + 6w = 85$ $15c + 8w = 85 \quad \text{(Equation 3)}$

Equation 2: $9(2c) + 1w + 4c + 5(2w) = 121$ $18c + w + 4c + 10w = 121$ $22c + 11w = 121 \quad \text{(Equation 4)}$

Step 4: Set up the system of linear equations

Now we have the system of equations:

15c + 8w &= 85 \quad \text{(Equation 3)} \\ 22c + 11w &= 121 \quad \text{(Equation 4)} \end{aligned}$$ ### Step 5: Solve the system using Cramer's rule The system can be written in matrix form $$A \mathbf{x} = \mathbf{b}$$, where: $$A = \begin{pmatrix} 15 & 8 \\ 22 & 11 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} c \\ w \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 85 \\ 121 \end{pmatrix}$$ Using Cramer's rule: $$c = \frac{\text{det}(A_c)}{\text{det}(A)}, \quad w = \frac{\text{det}(A_w)}{\text{det}(A)}$$ where: $$A_c = \begin{pmatrix} 85 & 8 \\ 121 & 11 \end{pmatrix}, \quad A_w = \begin{pmatrix} 15 & 85 \\ 22 & 121 \end{pmatrix}$$ First, we calculate $$\text{det}(A)$$: $$\text{det}(A) = 15(11) - 8(22) = 165 - 176 = -11$$ Next, we calculate $$\text{det}(A_w)$$: $$\text{det}(A_w) = 15(121) - 22(85) = 1815 - 1870 = -55$$ Finally, using Cramer's rule: $$w = \frac{-55}{-11} = 5$$ Since $$h = 2w$$, the cost of one whiskey $$h$$ is: $$h = 2(5) = 10$$ ### Final Answer The cost of one whiskey is **10**.