Question

system is equivalent to

\\(\

$$\begin{cases} 3x^2 - 4y^2 = 25 \\\\ 12x^2 + 4y^2 = 22 \\end{cases}$$

\\)

\\(\

$$\begin{cases} 3x^2 - 4y^2 = 25 \\\\ -12x^2 + 4y^2 = 22 \\end{cases}$$

\\)

\\(\

$$\begin{cases} 6x^2 - 8y^2 = 25 \\\\ -6x^2 - 2y^2 = 11 \\end{cases}$$

\\)

\\(\

$$\begin{cases} 6x^2 - 8y^2 = 50 \\\\ -6x^2 - 2y^2 = 11 \\end{cases}$$

\\)

Explanation:

Identify the original system of equations

The top of the image shows the second equation of the original system:

$$ -6x^2 - 2y^2 = 11 $$

Looking at the options, the first equation in the original system must be:

$$ 3x^2 - 4y^2 = 25 $$

Thus, the original system is:

$$
LATEXBLOCK0
$$

Analyze the transformations in the options

• Option 1:

$$
LATEXBLOCK1
$$

The second equation is multiplied by \(-2\), which should yield \(12x^2 + 4y^2 = -22\), not \(22\).

• Option 2:

$$
LATEXBLOCK2
$$

The second equation is multiplied by \(2\), which should yield \(-12x^2 - 4y^2 = 22\), not \(-12x^2 + 4y^2 = 22\).

• Option 3:

$$
LATEXBLOCK3
$$

The first equation is multiplied by \(2\) on the left side, but the right side remains \(25\) instead of \(50\).

• Option 4:

$$
LATEXBLOCK4
$$

The first equation is multiplied by \(2\) on both sides: \(2(3x^2 - 4y^2) = 2(25) \implies 6x^2 - 8y^2 = 50\). The second equation remains unchanged.

Determine the correct equivalent system

The fourth option correctly multiplies the first equation by \(2\) while keeping the second equation unchanged, resulting in an equivalent system.

Answer:

• (A) \(

$$\begin{cases} 3x^2 - 4y^2 = 25 \\ 12x^2 + 4y^2 = 22 \end{cases}$$

\)

• (B) \(

$$\begin{cases} 3x^2 - 4y^2 = 25 \\ -12x^2 + 4y^2 = 22 \end{cases}$$

\)

• (C) \(

$$\begin{cases} 6x^2 - 8y^2 = 25 \\ -6x^2 - 2y^2 = 11 \end{cases}$$

\)

• **(D) \(

$$\begin{cases} 6x^2 - 8y^2 = 50 \\ -6x^2 - 2y^2 = 11 \end{cases}$$

\) (Correct answer)**