Question

to solve the system of equations below, zach isolated ( x^2 ) in the first equation and then substituted it into the second equation. what was the resulting equation?
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$$\begin{cases} x^2 + y^2 = 25 \\\\ \\dfrac{x^2}{16} - \\dfrac{y^2}{9} = 1 \\end{cases}$$

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\\( \boldsymbol{\text{a. } \dfrac{y^2 - 25}{16} - \dfrac{y^2}{9} = 1} \\)\\( \boldsymbol{\text{b. } \dfrac{25 - y^2}{16} - \dfrac{y^2}{9} = 1} \\)\\( \boldsymbol{\text{c. } \dfrac{x^2}{16} - \dfrac{y^2 - 25}{9} = 1} \\)\\( \boldsymbol{\text{d. } \dfrac{x^2}{16} - \dfrac{25 - y^2}{9} = 1} \\)

Explanation:

Step1: Isolate \( x^2 \) from the first equation

The first equation is \( x^2 + y^2 = 25 \). To isolate \( x^2 \), we subtract \( y^2 \) from both sides.
\[
x^2 = 25 - y^2
\]

Step2: Substitute \( x^2 \) into the second equation

The second equation is \( \frac{x^2}{16} - \frac{y^2}{9} = 1 \). Now we substitute \( x^2 = 25 - y^2 \) into this equation.
\[
\frac{25 - y^2}{16} - \frac{y^2}{9} = 1
\]

Answer:

B. \(\frac{25 - y^2}{16} - \frac{y^2}{9} = 1\)