To determine which system of equations is equivalent to the original system:
[tex]\[
\left\{\begin{array}{l}
5 x^2 + 6 y^2 = 50 \\
7 x^2 + 2 y^2 = 10
\end{array}\right.
\][/tex]
we transform the original equations to create systems of equations and compare them with the given options.
First, observe the transformations necessary to achieve certain forms.
Transforming the first equation:
We start with:
[tex]\[ 5x^2 + 6y^2 = 50 \][/tex]
To derive an equivalent equation with [tex]\(35x^2 + 42y^2\)[/tex] on the left side, multiply every term by 7:
[tex]\[ 7 \cdot (5x^2 + 6y^2) = 7 \cdot 50 \][/tex]
[tex]\[ 35x^2 + 42y^2 = 350 \][/tex]
We get:
[tex]\[ 35x^2 + 42y^2 = 350 \][/tex]
Transforming the second equation:
We start with:
[tex]\[ 7x^2 + 2y^2 = 10 \][/tex]
To derive an equivalent equation with [tex]\(-35x^2 - 10y^2\)[/tex] on the left side, multiply every term by -5:
[tex]\[ -5 \cdot (7x^2 + 2y^2) = -5 \cdot 10 \][/tex]
[tex]\[ -35x^2 - 10y^2 = -50 \][/tex]
We get:
[tex]\[ -35x^2 - 10y^2 = -50 \][/tex]
Thus, our transformed system of equations is:
[tex]\[
\left\{\begin{array}{l}
35 x^2 + 42 y^2 = 350 \\
-35 x^2 - 10 y^2 = -50
\end{array}\right.
\][/tex]
Now, let's compare this with the given options:
1. [tex]\[
\left\{\begin{array}{r}
5 x^2 + 6 y^2 = 50 \\
-21 x^2 - 6 y^2 = 10
\end{array}\right.
\][/tex]
This does not match as it hasn't applied full transformations required.
2. [tex]\[
\left\{\begin{aligned}
5 x^2 + 6 y^2 & =50 \\
-21 x^2 - 6 y^2 & =30
\end{aligned}\right.
\][/tex]
This also doesn't match.
3. [tex]\[
\left\{\begin{aligned}
35 x^2 + 42 y^2 & =250 \\
-35 x^2 - 10 y^2 & =-50
\end{aligned}\right.
\][/tex]
The equation [tex]\(35x^2 + 42y^2 = 250\)[/tex] does not match; the right side should be 350.
4. [tex]\[
\left\{\begin{array}{r}
35 x^2 + 42 y^2 = 350 \\
-35 x^2 - 10 y^2 = -50
\end{array}\right.
\][/tex]
This system matches perfectly.
Hence, the system equivalent to the original is:
[tex]\[
\left\{\begin{array}{r}
35 x^2 + 42 y^2 = 350 \\
-35 x^2 - 10 y^2 = -50
\end{array}\right.
\][/tex]
