High School

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

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

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

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

Answer :

To determine which system is equivalent to the given system of equations:

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

We need to find an equivalent system from the options provided by scaling and combining these original equations. Let's examine each option:

### Option 1:
1. [tex]\(5x^2 + 6y^2 = 50\)[/tex]
2. [tex]\(-21x^2 - 6y^2 = 10\)[/tex]

This system does not match the original equation when scaled or combined.

### Option 2:
1. [tex]\(5x^2 + 6y^2 = 50\)[/tex]
2. [tex]\(-21x^2 - 6y^2 = 30\)[/tex]

This system does not match the original equation when scaled or combined.

### Option 3:
1. [tex]\(35x^2 + 42y^2 = 250\)[/tex]
2. [tex]\(-35x^2 - 10y^2 = -50\)[/tex]

To check if this system could be equivalent, let's multiply the original equations by a suitable factor to see if they align.

- Multiply the first original equation by 7:
- [tex]\(7(5x^2 + 6y^2 = 50)\)[/tex] gives [tex]\(35x^2 + 42y^2 = 350\)[/tex]

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

The attempt to check if the resulting equations can be equivalent does not lead to matching with Option 3 as shown in the calculations, so this is not equivalent.

### Option 4:
1. [tex]\(35x^2 + 42y^2 = 350\)[/tex]
2. [tex]\(-35x^2 - 10y^2 = -50\)[/tex]

This attempt also does not match when scaling the equations from the original to fit this system.

Conclusion: None of the options provided forms an equivalent system to the original set of equations. Therefore, the correct selection aligns with no equivalent system present in the options given.