In the inn U Temného hvozdu, giant twins Pravdoslav and Křivomil serve the guests. Pravdoslav is honest and always charges accurately, while Křivomil is dishonest and always adds two krejcars to the price of each pie and each jug of mead. One day, seven dwarfs visited this inn and sat at two tables. The dwarfs paid the same amount for four pies from one giant as they did for three jugs of mead from the other. Another time, the dwarfs paid 14 krejcars more for four jugs of mead from Křivomil than for three pies from Pravdoslav.

Determine how much one pie and one jug of mead cost at Pravdoslav. All prices are in whole krejcars and did not change between the two visits.

(M. Petrová, M. Dillingerová)
The actual price (the price at Pravdoslav) of one jug of mead we denote as $m$ and the actual price of one cake we denote as $k$. During their first visit, the dwarves could pay for cakes at Pravdoslav and mead at Křivomil, or vice versa. This corresponds to two different expressions:

$$
4 k=3(m+2)
$$

or

$$
4(k+2)=3 m \text {. }
$$

According to the information about the second visit, we know that

$$
4(m+2)=3 k+14
$$

The solution to the system of equations (1a) and (2) is the pair

$$
k=6, m=6
$$

the solution to the system of equations (1b) and (2) is the pair

$$
k=-2, m=0 .
$$

Given that the price of any item is always positive, only the first option is valid: both the cake and the jug of mead cost 6 kreuzers at Pravdoslav.

Evaluation. 1 point for formulating each of the conditions (1a), (1b), and (2); 1 point for solving each of the systems of equations; 1 point for the result.