On the dream market, the Sphinx offered a traveler four dreams for seven illusions, two slippers, and one nightmare. To another traveler, the same Sphinx offered seven dreams for four illusions, four slippers, and two nightmares. The Sphinx measures all travelers equally.

How many illusions did one dream cost?

(K. Pazourek)

Hint. Express the relationships from the problem using variables, dream a little, and look for further relationships.
Let's denote the price of a dream, illusion, slumber, and nightmare as $s, i, \check{s}$, and $m$ respectively. Then, according to the problem, we have:

$$
4 s=7 i+2 \check{s}+m, \quad 7 s=4 i+4 \check{s}+2 m .
$$

The number of slumbers and nightmares in the second equation is twice that in the first equation. By expressing these commodities from the first equation and substituting them into the second, we get the relationship between dreams and illusions:

$$
\begin{aligned}
4 s-7 i=2 \check{s}+m, \quad 7 s & =4 i+2(2 \check{s}+m), \\
7 s & =4 i+8 s-14 i, \\
s & =10 i .
\end{aligned}
$$

One dream costs ten illusions.

Notes. From the problem, it is not possible to determine the remaining unknowns, but only to derive further relationships between them, or more precisely, to express the given relationships differently. For the given equations, any manipulation (applied simultaneously to both sides of the equations) will yield another valid equation. When solving linear equations, we can eliminate unknowns using appropriate linear combinations of the given equations. Our case $(*)$ is special in that two of the four unknowns can be eliminated at once: the difference between the second equation and twice the first gives

$$
\begin{aligned}
8 s-7 s & =14 i+4 \check{s}+2 m-4 i-4 \check{s}-2 m \\
s & =10 i
\end{aligned}
$$

This is just another way of grasping the same insight as in the previous solution.

The system of equations $(*)$ has infinitely many solutions, which can be expressed, for example, as

$$
s=10 i, \quad m=33 i-2 \check{s},
$$

where $i$ and $\check{s}$ can be any numbers. In particular, the quadruple $i=0, s=0, \check{s}=0, m=0$ is a solution, and all solutions are linear combinations of the quadruples $i=0, s=0, \check{s}=1, m=-2$ and $i=1$, $s=10, \check{s}=0, m=33$.

For any solution of the system $(*)$ that is experimentally discovered, it will certainly hold that $s=10 i$. Generalizations of the type

$$
, i=1, s=10, \check{s}=0, m=33 \text { is a solution, thus } s=10 i \text { " }
$$

cannot be considered a satisfactory solution to the problem without additional explanation.