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]
