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Answer :
To determine which system is equivalent to the given system of equations, we start with the original system:
1) [tex]\(5x^2 + 6y^2 = 50\)[/tex]
2) [tex]\(7x^2 + 2y^2 = 10\)[/tex]
We will try to manipulate these equations to see if they can match any of the given options.
### Step-by-step Solution:
Step 1: Equalize the coefficients of [tex]\(x^2\)[/tex].
To make the coefficients of [tex]\(x^2\)[/tex] the same in both original equations, we can multiply the entire first equation by 7 and the entire second equation by 5:
- Multiply the first equation by 7:
[tex]\[
7(5x^2 + 6y^2) = 7 \times 50
\][/tex]
Which simplifies to:
[tex]\[
35x^2 + 42y^2 = 350
\][/tex]
- Multiply the second equation by 5:
[tex]\[
5(7x^2 + 2y^2) = 5 \times 10
\][/tex]
Which simplifies to:
[tex]\[
35x^2 + 10y^2 = 50
\][/tex]
Step 2: Check equivalence with the given options.
The transformed equations are:
- [tex]\[35x^2 + 42y^2 = 350\][/tex]
- [tex]\[35x^2 + 10y^2 = 50\][/tex]
But notice that these are not exactly what we see in the options. So, let's look for negative sign counterparts that are equivalent transformations:
If we instead write:
- [tex]\[35x^2 + 42y^2 = 350\][/tex]
- [tex]\[ -35x^2 - 10y^2 = -50\][/tex]
This form fits with the fourth option:
[tex]\[
\left\{\begin{aligned} 35x^2+42y^2 & =350 \\ -35x^2-10y^2 & =-50\end{aligned}\right.
\][/tex]
This configuration results in equivalent values when manipulated. Therefore, the system that matches the transformations is:
[tex]\[
\left\{\begin{aligned} 35x^2+42y^2 & =350 \\ -35x^2-10y^2 & =-50\end{aligned}\right.
\][/tex]
The fourth option in the list given matches this, making it the correct equivalent system.
1) [tex]\(5x^2 + 6y^2 = 50\)[/tex]
2) [tex]\(7x^2 + 2y^2 = 10\)[/tex]
We will try to manipulate these equations to see if they can match any of the given options.
### Step-by-step Solution:
Step 1: Equalize the coefficients of [tex]\(x^2\)[/tex].
To make the coefficients of [tex]\(x^2\)[/tex] the same in both original equations, we can multiply the entire first equation by 7 and the entire second equation by 5:
- Multiply the first equation by 7:
[tex]\[
7(5x^2 + 6y^2) = 7 \times 50
\][/tex]
Which simplifies to:
[tex]\[
35x^2 + 42y^2 = 350
\][/tex]
- Multiply the second equation by 5:
[tex]\[
5(7x^2 + 2y^2) = 5 \times 10
\][/tex]
Which simplifies to:
[tex]\[
35x^2 + 10y^2 = 50
\][/tex]
Step 2: Check equivalence with the given options.
The transformed equations are:
- [tex]\[35x^2 + 42y^2 = 350\][/tex]
- [tex]\[35x^2 + 10y^2 = 50\][/tex]
But notice that these are not exactly what we see in the options. So, let's look for negative sign counterparts that are equivalent transformations:
If we instead write:
- [tex]\[35x^2 + 42y^2 = 350\][/tex]
- [tex]\[ -35x^2 - 10y^2 = -50\][/tex]
This form fits with the fourth option:
[tex]\[
\left\{\begin{aligned} 35x^2+42y^2 & =350 \\ -35x^2-10y^2 & =-50\end{aligned}\right.
\][/tex]
This configuration results in equivalent values when manipulated. Therefore, the system that matches the transformations is:
[tex]\[
\left\{\begin{aligned} 35x^2+42y^2 & =350 \\ -35x^2-10y^2 & =-50\end{aligned}\right.
\][/tex]
The fourth option in the list given matches this, making it the correct equivalent system.
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