Which system is equivalent to [tex]\left\{\begin{array}{l}3 x^2-4 y^2=25 \\ -6 x^2-2 y^2=11\end{array}\right.[/tex]?

A. [tex]\left\{\begin{array}{l}3 x^2-4 y^2=25 \\ 12 x^2+4 y^2=22\end{array}\right.[/tex]

B. [tex]\left\{\begin{aligned} 3 x^2-4 y^2 & =25 \\ -12 x^2+4 y^2 & =22\end{aligned}\right.[/tex]

C. [tex]\left\{\begin{array}{r}6 x^2-8 y^2=25 \\ -6 x^2-2 y^2=11\end{array}\right.[/tex]

D. [tex]\left\{\begin{array}{r}6 x^2-8 y^2=50 \\ -6 x^2-2 y^2=11\end{array}\right.[/tex]



Answer :

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]