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

Answer :

Sure! Let's solve the equation step by step to isolate [tex]\( y^2 \)[/tex] in the equation [tex]\( 4x^2 + 25y^2 = 100 \)[/tex].

### Step-by-step Solution:

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

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

3. Solve for [tex]\( y^2 \)[/tex]:
Now, we need to divide every term by 25 to get [tex]\( y^2 \)[/tex] by itself:
[tex]\[
y^2 = \frac{100 - 4x^2}{25}
\][/tex]

4. Simplify the expression:
Divide each term in the numerator by 25:
[tex]\[
y^2 = \frac{100}{25} - \frac{4x^2}{25}
\][/tex]

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

Therefore, the correct answer is:
- D. [tex]\( y^2 = 4 - \frac{4}{25}x^2 \)[/tex]

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