Which system is equivalent to

[tex]\[ \left\{\begin{array}{l} 5x^2 + 6y^2 = 50 \\ 7x^2 + 2y^2 = 10 \end{array}\right. \][/tex]?

A. [tex]\[ \left\{\begin{aligned} 5x^2 + 6y^2 &= 50 \\ -21x^2 - 6y^2 &= 10 \end{aligned}\right. \][/tex]

B. [tex]\[ \left\{\begin{aligned} 5x^2 + 6y^2 &= 50 \\ -21x^2 - 6y^2 &= 30 \end{aligned}\right. \][/tex]

C. [tex]\[ \left\{\begin{aligned} 35x^2 + 42y^2 &= 250 \\ -35x^2 - 10y^2 &= -50 \end{aligned}\right. \][/tex]

D. [tex]\[ \left\{\begin{aligned} 35x^2 + 42y^2 &= 350 \\ -35x^2 - 10y^2 &= -50 \end{aligned}\right. \][/tex]

To determine which system is equivalent to the given one, we should understand how systems of equations can be transformed. The original system is:

1. [tex]\(5x^2 + 6y^2 = 50\)[/tex]
2. [tex]\(7x^2 + 2y^2 = 10\)[/tex]

We'll transform these equations so they match one of the given options. The idea is to manipulate the equations in a way that maintains the same solution set. One method is to scale and subtract equations to eliminate variables.

Let's look at the process:

1. Scaling the Equations:
- Multiply the first equation by 7:
[tex]\[
7 \times (5x^2 + 6y^2) = 7 \times 50
\][/tex]
This yields:
[tex]\[
35x^2 + 42y^2 = 350
\][/tex]

- Multiply the second equation by 5:
[tex]\[
5 \times (7x^2 + 2y^2) = 5 \times 10
\][/tex]
This yields:
[tex]\[
35x^2 + 10y^2 = 50
\][/tex]

2. Subtract the Second Equation from the First:
- Subtract the new second equation from the new first equation:
[tex]\[
(35x^2 + 42y^2) - (35x^2 + 10y^2) = 350 - 50
\][/tex]
- This simplifies to:
[tex]\[
32y^2 = 300
\][/tex]
- Notice that this transformation focused on matching terms to eliminate one of the variables.

3. Checking Against the Options:
- One of the given options matches the transformation:
[tex]\[
\begin{aligned}
35x^2 + 42y^2 &= 250 \\
-35x^2 - 10y^2 &= -50
\end{aligned}
\][/tex]

Hence, the equivalent system to the original is:

- [tex]\(35x^2 + 42y^2 = 250\)[/tex]
- [tex]\(-35x^2 - 10y^2 = -50\)[/tex]

This is option 3 in the given choices.