To determine which system of equations is equivalent to the given system
[tex]\[
\left\{
\begin{array}{c}
3x^2 - 4y^2 = 25 \\
-6x^2 - 2y^2 = 11
\end{array}
\right.
\][/tex]
we need to manipulate the original equations and compare them to the options provided.
First, let's consider the original system of equations:
1. [tex]\( 3x^2 - 4y^2 = 25 \)[/tex]
2. [tex]\( -6x^2 - 2y^2 = 11 \)[/tex]
To compare these equations to the options given, let's manipulate the first equation to have a similar form to the second equation in absolute magnitude:
Multiply the first equation by 2:
[tex]\[
2 \times (3x^2 - 4y^2) = 2 \times 25
\][/tex]
This gives us:
[tex]\[
6x^2 - 8y^2 = 50
\][/tex]
Now we need to compare the resulting equation [tex]\( 6x^2 - 8y^2 = 50 \)[/tex] and the second equation [tex]\( -6x^2 - 2y^2 = 11 \)[/tex] to find the matching system in the given options.
Let's examine each option:
1. [tex]\(\left\{
\begin{array}{r}
3x^2 - 4y^2 = 25 \\
12x^2 + 4y^2 = 22
\end{array}
\right.\)[/tex]
This is not equivalent as the second equation doesn't match the manipulated form of the first system.
2. [tex]\(\left\{
\begin{array}{r}
3x^2 - 4y^2 = 25 \\
-12x^2 + 4y^2 = 22
\end{array}
\right.\)[/tex]
This is not equivalent because the coefficients don’t align after considering the multiplication transformation.
3. [tex]\(\left\{
\begin{array}{l}
6x^2 - 8y^2 = 25 \\
-6x^2 - 2y^2 = 11
\end{array}
\right.\)[/tex]
This is not equivalent; the first equation is incorrect after multiplication.
4. [tex]\(\left\{
\begin{array}{l}
6x^2 - 8y^2 = 50 \\
-6x^2 - 2y^2 = 11
\end{array}
\right.\)[/tex]
This is equivalent because both equations match the manipulated forms of the original system.
Therefore, the equivalent system is:
[tex]\[
\left\{
\begin{array}{l}
6x^2 - 8y^2 = 50 \\
-6x^2 - 2y^2 = 11
\end{array}
\right.
\][/tex]
Thus, the correct choice is:
4. [tex]\(\left\{
\begin{array}{l}
6x^2 - 8y^2 = 50 \\
-6x^2 - 2y^2 = 11
\end{array}
\right.\)[/tex]
