High School

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

Answer :

We start with the equation

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

Step 1. Isolate the [tex]$y^2$[/tex] term by subtracting [tex]$4x^2$[/tex] from both sides:

[tex]$$
25y^2 = 100 - 4x^2.
$$[/tex]

Step 2. Divide both sides of the equation by [tex]$25$[/tex] to solve for [tex]$y^2$[/tex]:

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

Step 3. Simplify the fraction by breaking it into two parts:

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

Since [tex]$\frac{100}{25} = 4$[/tex], we have:

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

This shows that the correct answer is:

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

Thus, the answer is option A.

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