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Answer :
To determine which system of equations is equivalent to the given system, we need to manipulate the original equations so they resemble one of the choices provided.
We start with the original system:
1. [tex]\( 5x^2 + 6y^2 = 50 \)[/tex]
2. [tex]\( 7x^2 + 2y^2 = 10 \)[/tex]
The goal is to manipulate these equations in such a way that they match one of the given systems. One effective method to achieve this is by trying to eliminate one of the terms in both equations. Here's how we can do this:
1. Manipulation of the first equation:
[tex]\[
5x^2 + 6y^2 = 50
\][/tex]
Multiply the entire equation by 7 to help eliminate [tex]\(x^2\)[/tex] in the subsequent step:
[tex]\[
35x^2 + 42y^2 = 350
\][/tex]
2. Manipulation of the second equation:
[tex]\[
7x^2 + 2y^2 = 10
\][/tex]
Multiply the entire equation by [tex]\(-5\)[/tex] to help eliminate [tex]\(x^2\)[/tex]:
[tex]\[
-35x^2 - 10y^2 = -50
\][/tex]
Combining these manipulated equations, we get the following system:
- [tex]\(35x^2 + 42y^2 = 350\)[/tex]
- [tex]\(-35x^2 - 10y^2 = -50\)[/tex]
When we compare this manipulated system with the given choices, we find that it matches the fourth choice exactly:
[tex]\[ \left\{\begin{array}{l}35 x^2+42 y^2=350 \\ -35 x^2-10 y^2=-50\end{array}\right. \][/tex]
Therefore, the equivalent system of equations is the fourth option:
[tex]\[ \left\{\begin{array}{l}35 x^2+42 y^2=350 \\ -35 x^2-10 y^2=-50\end{array}\right. \][/tex]
We start with the original system:
1. [tex]\( 5x^2 + 6y^2 = 50 \)[/tex]
2. [tex]\( 7x^2 + 2y^2 = 10 \)[/tex]
The goal is to manipulate these equations in such a way that they match one of the given systems. One effective method to achieve this is by trying to eliminate one of the terms in both equations. Here's how we can do this:
1. Manipulation of the first equation:
[tex]\[
5x^2 + 6y^2 = 50
\][/tex]
Multiply the entire equation by 7 to help eliminate [tex]\(x^2\)[/tex] in the subsequent step:
[tex]\[
35x^2 + 42y^2 = 350
\][/tex]
2. Manipulation of the second equation:
[tex]\[
7x^2 + 2y^2 = 10
\][/tex]
Multiply the entire equation by [tex]\(-5\)[/tex] to help eliminate [tex]\(x^2\)[/tex]:
[tex]\[
-35x^2 - 10y^2 = -50
\][/tex]
Combining these manipulated equations, we get the following system:
- [tex]\(35x^2 + 42y^2 = 350\)[/tex]
- [tex]\(-35x^2 - 10y^2 = -50\)[/tex]
When we compare this manipulated system with the given choices, we find that it matches the fourth choice exactly:
[tex]\[ \left\{\begin{array}{l}35 x^2+42 y^2=350 \\ -35 x^2-10 y^2=-50\end{array}\right. \][/tex]
Therefore, the equivalent system of equations is the fourth option:
[tex]\[ \left\{\begin{array}{l}35 x^2+42 y^2=350 \\ -35 x^2-10 y^2=-50\end{array}\right. \][/tex]
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