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What is the result of isolating [tex]$y^2$[/tex] in the equation below?

[tex]4 x^2 + 25 y^2 = 100[/tex]

A. [tex]y^2 = 100 - \frac{4}{25} x^2[/tex]

B. [tex]y^2 = 25 - \frac{4}{25} x^2[/tex]

C. [tex]y^2 = 4 - \frac{4}{25} x^2[/tex]

D. [tex]y^2 = 100 - 4 x^2[/tex]

Answer :

To isolate [tex]\( y^2 \)[/tex] in the given equation [tex]\( 4x^2 + 25y^2 = 100 \)[/tex], follow these steps:

1. Start with the equation:
[tex]\[ 4x^2 + 25y^2 = 100 \][/tex]

2. Subtract [tex]\( 4x^2 \)[/tex] from both sides to begin isolating [tex]\( y^2 \)[/tex]:
[tex]\[ 25y^2 = 100 - 4x^2 \][/tex]

3. Now, divide both sides by 25 to solve for [tex]\( y^2 \)[/tex]:
[tex]\[ y^2 = \frac{100 - 4x^2}{25} \][/tex]

4. Simplify the right-hand side of the equation.
Split the fraction:
[tex]\[ y^2 = \frac{100}{25} - \frac{4x^2}{25} \][/tex]

5. Calculate the simplified terms:
[tex]\[ y^2 = 4 - \frac{4}{25}x^2 \][/tex]

So, the correct option is:
[tex]\[ \boxed{C \,\, y^2 = 4 - \frac{4}{25} x^2} \][/tex]

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