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

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

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

Answer :

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

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

2. Move the [tex]\( 4x^2 \)[/tex] term to the other side of the equation:
Subtract [tex]\( 4x^2 \)[/tex] from both sides to isolate the terms with [tex]\( y^2 \)[/tex].
[tex]\[
25y^2 = 100 - 4x^2
\][/tex]

3. Solve for [tex]\( y^2 \)[/tex]:
To isolate [tex]\( y^2 \)[/tex], divide every term by 25.
[tex]\[
y^2 = \frac{100 - 4x^2}{25}
\][/tex]

4. Simplify the right side:
Break down the fraction:
[tex]\[
y^2 = \frac{100}{25} - \frac{4x^2}{25}
\][/tex]
Calculate [tex]\( \frac{100}{25} = 4 \)[/tex] and write the expression:
[tex]\[
y^2 = 4 - \frac{4}{25}x^2
\][/tex]

After following these steps, we find that:
[tex]\[
y^2 = 4 - \frac{4}{25}x^2
\][/tex]

So, the correct result for isolating [tex]\( y^2 \)[/tex] is [tex]\( y^2 = 4 - \frac{4}{25}x^2 \)[/tex].

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