7. (6 points) The triples of numbers $x, y, z$ and $a, b, c$ satisfy the system of equations

$$
\left\{\begin{array}{c}
12 x^{2}+4 y^{2}+9 z^{2}=20 \\
9 a^{2}+b^{2}+3 c^{2}=25
\end{array}\right.
$$

Within what limits can the expression $2cx-2ay+bz$ vary?
## Solution:

If we consider the equation of the system in the language of vector algebra, then the expression $12 x^{2}+4 y^{2}+9 z^{2}=20$ corresponds to

$$
\left\{\begin{array}{c}
|\vec{u}|=\sqrt{12 x^{2}+4 y^{2}+9 z^{2}}=2 \sqrt{5} \\
\vec{u}=\{2 \sqrt{3} x ; 2 y ; 3 z\}
\end{array}\right.
$$

The expression $9 a^{2}+b^{2}+3 c^{2}=25$ corresponds to

$$
\left\{\begin{array}{c}
|\vec{v}|=\sqrt{9 a^{2}+b^{2}+3 c^{2}}=5 \\
\overrightarrow{\mathrm{v}}=\{\sqrt{3} c ;-3 a ; b\}
\end{array}\right.
$$

Then the dot product of vectors $\vec{u}$ and $\vec{v}$ is $(\vec{u} \cdot \vec{v})=|\vec{u}| \cdot|\vec{v}| \cos \varphi=6 c x-6 a y+3 b z=3(2 c x-2 a y+b z)$

Since $\cos \varphi \in[-1,1]$, then $2 c x-2 a y+b z=\frac{1}{3}(\vec{u} \cdot \vec{v})=\frac{10}{3} \sqrt{5} \cos \varphi$.

Answer: $\left[-\frac{10}{3} \sqrt{5} ; \frac{10}{3} \sqrt{5}\right]$