Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Explore in-depth answers to your questions from a knowledgeable community of experts across different fields. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
To determine which system is equivalent to the given system of equations:
[tex]\[ \begin{cases} 5x^2 + 6y^2 = 50 \\ 7x^2 + 2y^2 = 10 \end{cases} \][/tex]
We need to verify each of the four provided systems step by step.
Step 1: Verify the first system:
[tex]\[ \begin{cases} 5x^2 + 6y^2 = 50 \\ -21x^2 - 6y^2 = 10 \end{cases} \][/tex]
This system can be rewritten for comparison:
- The first equation remains: \( 5x^2 + 6y^2 = 50 \).
- Simplify the second equation to identify if it aligns with the given original equations:
[tex]\[ -21x^2 - 6y^2 = 10 \Rightarrow \text{not equivalent to } 7x^2 + 2y^2 = 10 \][/tex]
This system is NOT equivalent.
Step 2: Verify the second system:
[tex]\[ \begin{cases} 5x^2 + 6y^2 = 50 \\ -21x^2 - 6y^2 = 30 \end{cases} \][/tex]
This system can be rewritten for comparison:
- The first equation remains: \( 5x^2 + 6y^2 = 50 \).
- Simplify the second equation to identify if it aligns with the given original equation:
[tex]\[ -21x^2 - 6y^2 = 30 \Rightarrow \text{not equivalent to } 7x^2 + 2y^2 = 10 \][/tex]
This system is NOT equivalent.
Step 3: Verify the third system:
[tex]\[ \begin{cases} 35x^2 + 42y^2 = 250 \\ -35x^2 - 10y^2 = -50 \end{cases} \][/tex]
This system can be rewritten for comparison:
- The first equation from the given system, when multiplied by 7, becomes:
[tex]\[ 7 \cdot (5x^2 + 6y^2) = 7 \cdot 50 \Rightarrow 35x^2 + 42y^2 = 350 \text{ (not 250)} \][/tex]
This system is NOT equivalent.
Step 4: Verify the fourth system:
[tex]\[ \begin{cases} 35x^2 + 42y^2 = 350 \\ -35x^2 - 10y^2 = -50 \end{cases} \][/tex]
This system can be rewritten for comparison:
- The first equation from the given system, when multiplied by 7, becomes:
[tex]\[ 7 \cdot (5x^2 + 6y^2) = 7 \cdot 50 \Rightarrow 35x^2 + 42y^2 = 350 \][/tex]
This is equivalent.
- The second equation from the given system, when multiplied by 5, becomes:
[tex]\[ 5 \cdot (7x^2 + 2y^2) = 5 \cdot 10 \Rightarrow 35x^2 + 10y^2 = 50 \][/tex]
When rearranged to an equivalent form with a negative sign, it is:
[tex]\[ -35x^2 - 10y^2 = -50 \][/tex]
Which matches the second equation in this system.
Thus, the fourth system is equivalent.
Conclusion:
The system equivalent to the given one is:
[tex]\[ \left\{\begin{array}{l} 35x^2 + 42y^2 = 350 \\ -35x^2 - 10y^2 = -50 \end{array}\right. \][/tex]
[tex]\[ \begin{cases} 5x^2 + 6y^2 = 50 \\ 7x^2 + 2y^2 = 10 \end{cases} \][/tex]
We need to verify each of the four provided systems step by step.
Step 1: Verify the first system:
[tex]\[ \begin{cases} 5x^2 + 6y^2 = 50 \\ -21x^2 - 6y^2 = 10 \end{cases} \][/tex]
This system can be rewritten for comparison:
- The first equation remains: \( 5x^2 + 6y^2 = 50 \).
- Simplify the second equation to identify if it aligns with the given original equations:
[tex]\[ -21x^2 - 6y^2 = 10 \Rightarrow \text{not equivalent to } 7x^2 + 2y^2 = 10 \][/tex]
This system is NOT equivalent.
Step 2: Verify the second system:
[tex]\[ \begin{cases} 5x^2 + 6y^2 = 50 \\ -21x^2 - 6y^2 = 30 \end{cases} \][/tex]
This system can be rewritten for comparison:
- The first equation remains: \( 5x^2 + 6y^2 = 50 \).
- Simplify the second equation to identify if it aligns with the given original equation:
[tex]\[ -21x^2 - 6y^2 = 30 \Rightarrow \text{not equivalent to } 7x^2 + 2y^2 = 10 \][/tex]
This system is NOT equivalent.
Step 3: Verify the third system:
[tex]\[ \begin{cases} 35x^2 + 42y^2 = 250 \\ -35x^2 - 10y^2 = -50 \end{cases} \][/tex]
This system can be rewritten for comparison:
- The first equation from the given system, when multiplied by 7, becomes:
[tex]\[ 7 \cdot (5x^2 + 6y^2) = 7 \cdot 50 \Rightarrow 35x^2 + 42y^2 = 350 \text{ (not 250)} \][/tex]
This system is NOT equivalent.
Step 4: Verify the fourth system:
[tex]\[ \begin{cases} 35x^2 + 42y^2 = 350 \\ -35x^2 - 10y^2 = -50 \end{cases} \][/tex]
This system can be rewritten for comparison:
- The first equation from the given system, when multiplied by 7, becomes:
[tex]\[ 7 \cdot (5x^2 + 6y^2) = 7 \cdot 50 \Rightarrow 35x^2 + 42y^2 = 350 \][/tex]
This is equivalent.
- The second equation from the given system, when multiplied by 5, becomes:
[tex]\[ 5 \cdot (7x^2 + 2y^2) = 5 \cdot 10 \Rightarrow 35x^2 + 10y^2 = 50 \][/tex]
When rearranged to an equivalent form with a negative sign, it is:
[tex]\[ -35x^2 - 10y^2 = -50 \][/tex]
Which matches the second equation in this system.
Thus, the fourth system is equivalent.
Conclusion:
The system equivalent to the given one is:
[tex]\[ \left\{\begin{array}{l} 35x^2 + 42y^2 = 350 \\ -35x^2 - 10y^2 = -50 \end{array}\right. \][/tex]
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.