Answer :
To find which system is equivalent to the given system:
[tex]\[
\left\{
\begin{array}{l}
5x^2 + 6y^2 = 50 \\
7x^2 + 2y^2 = 10
\end{array}
\right.
\][/tex]
we need to check each option to see which transformed system matches the original system of equations after possible manipulation.
Let's consider how these transformations could work:
### Original System:
1. [tex]\(5x^2 + 6y^2 = 50\)[/tex]
2. [tex]\(7x^2 + 2y^2 = 10\)[/tex]
We can multiply these equations to see how we might transform them:
### Transformations:
- Let's multiply the first equation by a number to make the coefficients match one of the options.
Multiply the first equation by 7:
1. [tex]\(7(5x^2 + 6y^2) = 7 \times 50\)[/tex]
This gives us:
[tex]\(35x^2 + 42y^2 = 350\)[/tex]
And multiply the entire second equation by [tex]\(-5\)[/tex]:
2. [tex]\(-5(7x^2 + 2y^2) = -5 \times 10\)[/tex]
This gives us:
[tex]\(-35x^2 - 10y^2 = -50\)[/tex]
### Check the Options:
Now we check which option matches this transformed system:
- 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]\[
\left\{
\begin{array}{r}
35x^2 + 42y^2 = 350 \\
-35x^2 - 10y^2 = -50
\end{array}
\right.
\][/tex]
From our transformed equations:
1. [tex]\(35x^2 + 42y^2 = 350\)[/tex]
2. [tex]\(-35x^2 - 10y^2 = -50\)[/tex]
We see that Option 4 matches our calculations:
[tex]\[
\left\{
\begin{array}{r}
35x^2 + 42y^2 = 350 \\
-35x^2 - 10y^2 = -50
\end{array}
\right.
\][/tex]
Thus, the correct equivalent system is:
Option 4
[tex]\[
\left\{
\begin{array}{l}
5x^2 + 6y^2 = 50 \\
7x^2 + 2y^2 = 10
\end{array}
\right.
\][/tex]
we need to check each option to see which transformed system matches the original system of equations after possible manipulation.
Let's consider how these transformations could work:
### Original System:
1. [tex]\(5x^2 + 6y^2 = 50\)[/tex]
2. [tex]\(7x^2 + 2y^2 = 10\)[/tex]
We can multiply these equations to see how we might transform them:
### Transformations:
- Let's multiply the first equation by a number to make the coefficients match one of the options.
Multiply the first equation by 7:
1. [tex]\(7(5x^2 + 6y^2) = 7 \times 50\)[/tex]
This gives us:
[tex]\(35x^2 + 42y^2 = 350\)[/tex]
And multiply the entire second equation by [tex]\(-5\)[/tex]:
2. [tex]\(-5(7x^2 + 2y^2) = -5 \times 10\)[/tex]
This gives us:
[tex]\(-35x^2 - 10y^2 = -50\)[/tex]
### Check the Options:
Now we check which option matches this transformed system:
- 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]\[
\left\{
\begin{array}{r}
35x^2 + 42y^2 = 350 \\
-35x^2 - 10y^2 = -50
\end{array}
\right.
\][/tex]
From our transformed equations:
1. [tex]\(35x^2 + 42y^2 = 350\)[/tex]
2. [tex]\(-35x^2 - 10y^2 = -50\)[/tex]
We see that Option 4 matches our calculations:
[tex]\[
\left\{
\begin{array}{r}
35x^2 + 42y^2 = 350 \\
-35x^2 - 10y^2 = -50
\end{array}
\right.
\][/tex]
Thus, the correct equivalent system is:
Option 4
