Answer :
To determine which system is equivalent to the given system of equations:
Original System:
1. [tex]\(5x^2 + 6y^2 = 50\)[/tex]
2. [tex]\(7x^2 + 2y^2 = 10\)[/tex]
We need to find an equivalent system from the options provided by scaling and combining these original equations. Let's examine each option:
### Option 1:
1. [tex]\(5x^2 + 6y^2 = 50\)[/tex]
2. [tex]\(-21x^2 - 6y^2 = 10\)[/tex]
This system does not match the original equation when scaled or combined.
### Option 2:
1. [tex]\(5x^2 + 6y^2 = 50\)[/tex]
2. [tex]\(-21x^2 - 6y^2 = 30\)[/tex]
This system does not match the original equation when scaled or combined.
### Option 3:
1. [tex]\(35x^2 + 42y^2 = 250\)[/tex]
2. [tex]\(-35x^2 - 10y^2 = -50\)[/tex]
To check if this system could be equivalent, let's multiply the original equations by a suitable factor to see if they align.
- Multiply the first original equation by 7:
- [tex]\(7(5x^2 + 6y^2 = 50)\)[/tex] gives [tex]\(35x^2 + 42y^2 = 350\)[/tex]
- Multiply the second original equation by 5:
- [tex]\(5(7x^2 + 2y^2 = 10)\)[/tex] gives [tex]\(35x^2 + 10y^2 = 50\)[/tex]
The attempt to check if the resulting equations can be equivalent does not lead to matching with Option 3 as shown in the calculations, so this is not equivalent.
### Option 4:
1. [tex]\(35x^2 + 42y^2 = 350\)[/tex]
2. [tex]\(-35x^2 - 10y^2 = -50\)[/tex]
This attempt also does not match when scaling the equations from the original to fit this system.
Conclusion: None of the options provided forms an equivalent system to the original set of equations. Therefore, the correct selection aligns with no equivalent system present in the options given.
Original System:
1. [tex]\(5x^2 + 6y^2 = 50\)[/tex]
2. [tex]\(7x^2 + 2y^2 = 10\)[/tex]
We need to find an equivalent system from the options provided by scaling and combining these original equations. Let's examine each option:
### Option 1:
1. [tex]\(5x^2 + 6y^2 = 50\)[/tex]
2. [tex]\(-21x^2 - 6y^2 = 10\)[/tex]
This system does not match the original equation when scaled or combined.
### Option 2:
1. [tex]\(5x^2 + 6y^2 = 50\)[/tex]
2. [tex]\(-21x^2 - 6y^2 = 30\)[/tex]
This system does not match the original equation when scaled or combined.
### Option 3:
1. [tex]\(35x^2 + 42y^2 = 250\)[/tex]
2. [tex]\(-35x^2 - 10y^2 = -50\)[/tex]
To check if this system could be equivalent, let's multiply the original equations by a suitable factor to see if they align.
- Multiply the first original equation by 7:
- [tex]\(7(5x^2 + 6y^2 = 50)\)[/tex] gives [tex]\(35x^2 + 42y^2 = 350\)[/tex]
- Multiply the second original equation by 5:
- [tex]\(5(7x^2 + 2y^2 = 10)\)[/tex] gives [tex]\(35x^2 + 10y^2 = 50\)[/tex]
The attempt to check if the resulting equations can be equivalent does not lead to matching with Option 3 as shown in the calculations, so this is not equivalent.
### Option 4:
1. [tex]\(35x^2 + 42y^2 = 350\)[/tex]
2. [tex]\(-35x^2 - 10y^2 = -50\)[/tex]
This attempt also does not match when scaling the equations from the original to fit this system.
Conclusion: None of the options provided forms an equivalent system to the original set of equations. Therefore, the correct selection aligns with no equivalent system present in the options given.
