Solve the following system of equations:

$$
\begin{gathered}
x u+y v+z w=u^{\prime} \\
x^{\prime} u+y^{\prime} v+z^{\prime} w=v^{\prime} \\
x^{\prime \prime} u+y^{\prime \prime} v+z^{\prime \prime} w=w^{\prime}
\end{gathered}
$$

where $x, y, z, x^{\prime}, \ldots, z^{\prime \prime}$ are to be calculated from the systems of equations given in the previous problem.
Before determining the values of $u$, $v$, and $w$, let's expand the value of $\Delta_{\alpha}$:

$$
\Delta_{\alpha}=\left|\begin{array}{ccc}
x & y & z \\
x^{\prime} & y^{\prime} & z^{\prime} \\
x^{\prime \prime} & y^{\prime \prime} & z^{\prime \prime}
\end{array}\right|=x y^{\prime} z^{\prime \prime}+y z^{\prime} x^{\prime \prime}+z x^{\prime} y^{\prime \prime}-z y^{\prime} x^{\prime \prime}-x z^{\prime} y^{\prime \prime}-y x^{\prime} z^{\prime \prime}
$$

By substituting the values of $x, x^{\prime}, x^{\prime \prime}, y \ldots z^{\prime \prime}$ and performing the specified operations, we get:

$$
\Delta_{\alpha}=\frac{\left(1+r^{2}+p^{2}+q^{2}\right)^{3}}{\left(1+r^{2}+p^{2}+q^{2}\right)^{3}}=\frac{\Delta^{3}}{\Delta^{3}}=1
$$

Thus,

$$
u=\left|\begin{array}{ccc}
u^{\prime} & y & z \\
v^{\prime} & y^{\prime} & z^{\prime} \\
w^{\prime} & y^{\prime \prime} & z^{\prime \prime}
\end{array}\right|=u^{\prime} y^{\prime} z^{\prime \prime}+y z^{\prime} w^{\prime \prime}+z v^{\prime} y^{\prime \prime}-z y^{\prime} w^{\prime}-u^{\prime} z^{\prime} y^{\prime \prime}-y v^{\prime} z^{\prime \prime}
$$

By substituting the appropriate values, performing the specified multiplications, and simplifying the resulting expression, we get:

$$
u=\frac{u^{\prime}\left(1+p^{2}-r^{2}-q^{2}\right)+2 v^{\prime}(p q-r)+2 w^{\prime}(r p+q)}{1+r^{2}+p^{2}+q^{2}}
$$

Similarly, we obtain:

$$
v=\frac{2 u^{\prime}(r+p q)+v^{\prime}\left(1-r^{2}-p^{2}+q^{2}\right)+2 w^{\prime}(r q-p)}{1+r^{2}+p^{2}+q^{2}}
$$

and

$$
w=\frac{2 u^{\prime}(r p-q)+2 v^{\prime}(p+r q)+w^{\prime}\left(1+r^{2}-p^{2}-q^{2}\right)}{1+r^{2}+p^{2}+q^{2}}
$$

or

$$
\begin{aligned}
& u=u^{\prime} x+v^{\prime} x^{\prime}+w^{\prime} x^{\prime \prime} \\
& v=u^{\prime} y+v^{\prime} y^{\prime}+w^{\prime} y^{\prime \prime}
\end{aligned}
$$

and

$$
w=u^{\prime} z+v^{\prime} z^{\prime}+w^{\prime} z^{\prime \prime}
$$

(Krisztián György.)

The problem was also solved by: Freibauer E., Kiss A., Weisz Á.