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Which system is equivalent to

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

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

B.
\[
\left\{
\begin{aligned}
5x^2 + 8y^2 & = 50 \\
-21x^2 - 6y^2 & = 30
\end{aligned}
\right.
\]

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

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

Answer :

We start with the system

[tex]$$
\begin{cases}
5x^2 + 6y^2 = 50, \\
7x^2 + 2y^2 = 10.
\end{cases}
$$[/tex]

Our goal is to obtain an equivalent system whose equations are multiples of those in the original system. A good strategy is to eliminate one variable (or, in this case, the term [tex]$x^2$[/tex]) by scaling the equations appropriately.

1. Multiply the first equation by [tex]$7$[/tex] to obtain a new equation:

[tex]$$
7(5x^2 + 6y^2) = 7 \cdot 50,
$$[/tex]

which simplifies to

[tex]$$
35x^2 + 42y^2 = 350.
$$[/tex]

2. Multiply the second equation by [tex]$-5$[/tex] to obtain another new equation:

[tex]$$
-5(7x^2 + 2y^2) = -5 \cdot 10,
$$[/tex]

which simplifies to

[tex]$$
-35x^2 - 10y^2 = -50.
$$[/tex]

By multiplying the equations as shown, the system becomes

[tex]$$
\begin{cases}
35x^2 + 42y^2 = 350, \\
-35x^2 - 10y^2 = -50.
\end{cases}
$$[/tex]

This is equivalent to the original system. Comparing with the given options, the system that matches is:

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

Therefore, the correct equivalent system is the one corresponding to option 4.

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