To determine which system of equations is equivalent to the given system:
[tex]\[
\left\{
\begin{array}{l}
3x^2 - 4y^2 = 25 \\
-6x^2 - 2y^2 = 11
\end{array}
\right.
\][/tex]
we need to simplify each of the provided systems and see if they match the original system when simplified.
Option 1:
[tex]\[
\left\{
\begin{array}{l}
3x^2 - 4y^2 = 25 \\
12x^2 + 4y^2 = 22
\end{array}
\right.
\][/tex]
For the second equation, divide both sides by 2:
[tex]\[
12x^2 + 4y^2 = 22 \implies 6x^2 + 2y^2 = 11
\][/tex]
So Option 1 becomes:
[tex]\[
\left\{
\begin{array}{l}
3x^2 - 4y^2 = 25 \\
6x^2 + 2y^2 = 11
\end{array}
\right.
\][/tex]
This is not equivalent to the original system.
Option 2:
[tex]\[
\left\{
\begin{aligned}
3x^2 - 4y^2 & = 25 \\
-12x^2 + 4y^2 & = 22
\end{aligned}
\right.
\][/tex]
For the second equation, divide both sides by 2:
[tex]\[
-12x^2 + 4y^2 = 22 \implies -6x^2 + 2y^2 = 11
\][/tex]
So Option 2 becomes:
[tex]\[
\left\{
\begin{array}{l}
3x^2 - 4y^2 = 25 \\
-6x^2 + 2y^2 = 11
\end{array}
\right.
\][/tex]
This is not equivalent to the original system.
Option 3:
[tex]\[
\left\{
\begin{array}{l}
6x^2 - 8y^2 = 25 \\
-6x^2 - 2y^2 = 11
\end{array}
\right.
\][/tex]
Compare this directly to the original system. The first equation is not equivalent to [tex]\(3x^2 - 4y^2 = 25\)[/tex]. Even if divided by 2, the equality does not hold true. So, Option 3 is not equivalent.
Option 4:
[tex]\[
\left\{
\begin{array}{r}
6x^2 - 8y^2 = 50 \\
-6x^2 - 2y^2 = 11
\end{array}
\right.
\][/tex]
For the first equation, divide both sides by 2:
[tex]\[
6x^2 - 8y^2 = 50 \implies 3x^2 - 4y^2 = 25
\][/tex]
So Option 4 becomes:
[tex]\[
\left\{
\begin{array}{l}
3x^2 - 4y^2 = 25 \\
-6x^2 - 2y^2 = 11
\end{array}
\right.
\][/tex]
This system matches the original system exactly.
Conclusion:
The system that is equivalent to the given system is:
[tex]\[
\left\{
\begin{array}{r}
6x^2 - 8y^2 = 50 \\
-6x^2 - 2y^2 = 11
\end{array}
\right.
\][/tex]
So, the correct choice is:
[tex]\[
\boxed{4}
\][/tex]
