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Sagot :
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]
[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]
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