Answer :
To solve the problem of finding which system is equivalent to the given system of equations:
[tex]\[
\left\{
\begin{array}{l}
5x^2 + 6y^2 = 50 \\
7x^2 + 2y^2 = 10 \\
\end{array}
\right.
\][/tex]
we're going to manipulate these equations to see which option matches after manipulation.
### Step 1: Align the [tex]\( x^2 \)[/tex] terms for elimination
Multiply the first equation by 7 and the second equation by 5 to make the coefficients of [tex]\( x^2 \)[/tex] the same:
1. Multiply the first equation by 7:
[tex]\[
7(5x^2 + 6y^2 = 50) \implies 35x^2 + 42y^2 = 350
\][/tex]
2. Multiply the second equation by 5:
[tex]\[
5(7x^2 + 2y^2 = 10) \implies 35x^2 + 10y^2 = 50
\][/tex]
### Step 2: Subtract the second transformed equation from the first
Now subtract the newly formed second equation from the first to eliminate [tex]\( x^2 \)[/tex]:
[tex]\[
(35x^2 + 42y^2) - (35x^2 + 10y^2) = 350 - 50
\][/tex]
This simplifies to:
[tex]\[
42y^2 - 10y^2 = 300
\][/tex]
[tex]\[
32y^2 = 300
\][/tex]
The subtraction between left-hand sides doesn't fully match a typical representation here, indicating that the focus should remain on eliminating the [tex]\( x^2 \)[/tex] terms, allowing us to match these with a given option instead.
### Step 3: Check for equivalent options
Let's consider the options to see if any align:
- Option 1:
[tex]\[
\left\{
\begin{aligned}
5x^2 + 6y^2 & = 50 \\
-21x^2 - 6y^2 & = 10
\end{aligned}
\right.
\][/tex]
- Option 2:
[tex]\[
\left\{
\begin{aligned}
5x^2 + 6y^2 & = 50 \\
-21x^2 - 6y^2 & = 30
\end{aligned}
\right.
\][/tex]
- Option 3:
[tex]\[
\left\{
\begin{aligned}
35x^2 + 42y^2 & = 250 \\
-35x^2 - 10y^2 & = -50
\end{aligned}
\right.
\][/tex]
- Option 4:
[tex]\[
\left\{
\begin{aligned}
35x^2 + 42y^2 & = 350 \\
-35x^2 - 10y^2 & = -50
\end{aligned}
\right.
\][/tex]
Comparing these with the results from multiplying and aligning [tex]\( x^2 \)[/tex], we see that Option 4 is the one that matches:
[tex]\[
\left\{
\begin{aligned}
35x^2 + 42y^2 & = 350 \\
-35x^2 - 10y^2 & = -50
\end{aligned}
\right.
\][/tex]
Thus, the correct answer is the system:
[tex]\[
\left\{
\begin{aligned}
35x^2 + 42y^2 & = 350 \\
-35x^2 - 10y^2 & = -50
\end{aligned}
\right.
\][/tex]
[tex]\[
\left\{
\begin{array}{l}
5x^2 + 6y^2 = 50 \\
7x^2 + 2y^2 = 10 \\
\end{array}
\right.
\][/tex]
we're going to manipulate these equations to see which option matches after manipulation.
### Step 1: Align the [tex]\( x^2 \)[/tex] terms for elimination
Multiply the first equation by 7 and the second equation by 5 to make the coefficients of [tex]\( x^2 \)[/tex] the same:
1. Multiply the first equation by 7:
[tex]\[
7(5x^2 + 6y^2 = 50) \implies 35x^2 + 42y^2 = 350
\][/tex]
2. Multiply the second equation by 5:
[tex]\[
5(7x^2 + 2y^2 = 10) \implies 35x^2 + 10y^2 = 50
\][/tex]
### Step 2: Subtract the second transformed equation from the first
Now subtract the newly formed second equation from the first to eliminate [tex]\( x^2 \)[/tex]:
[tex]\[
(35x^2 + 42y^2) - (35x^2 + 10y^2) = 350 - 50
\][/tex]
This simplifies to:
[tex]\[
42y^2 - 10y^2 = 300
\][/tex]
[tex]\[
32y^2 = 300
\][/tex]
The subtraction between left-hand sides doesn't fully match a typical representation here, indicating that the focus should remain on eliminating the [tex]\( x^2 \)[/tex] terms, allowing us to match these with a given option instead.
### Step 3: Check for equivalent options
Let's consider the options to see if any align:
- Option 1:
[tex]\[
\left\{
\begin{aligned}
5x^2 + 6y^2 & = 50 \\
-21x^2 - 6y^2 & = 10
\end{aligned}
\right.
\][/tex]
- Option 2:
[tex]\[
\left\{
\begin{aligned}
5x^2 + 6y^2 & = 50 \\
-21x^2 - 6y^2 & = 30
\end{aligned}
\right.
\][/tex]
- Option 3:
[tex]\[
\left\{
\begin{aligned}
35x^2 + 42y^2 & = 250 \\
-35x^2 - 10y^2 & = -50
\end{aligned}
\right.
\][/tex]
- Option 4:
[tex]\[
\left\{
\begin{aligned}
35x^2 + 42y^2 & = 350 \\
-35x^2 - 10y^2 & = -50
\end{aligned}
\right.
\][/tex]
Comparing these with the results from multiplying and aligning [tex]\( x^2 \)[/tex], we see that Option 4 is the one that matches:
[tex]\[
\left\{
\begin{aligned}
35x^2 + 42y^2 & = 350 \\
-35x^2 - 10y^2 & = -50
\end{aligned}
\right.
\][/tex]
Thus, the correct answer is the system:
[tex]\[
\left\{
\begin{aligned}
35x^2 + 42y^2 & = 350 \\
-35x^2 - 10y^2 & = -50
\end{aligned}
\right.
\][/tex]
