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To solve the system of equations below, Zach isolated [tex]\(x^2\)[/tex] in the first equation and then substituted it into the second equation. What was the resulting equation?

[tex]\[
\left\{\begin{array}{l}
x^2 + y^2 = 25 \\
\frac{x^2}{16} - \frac{y^2}{9} = 1
\end{array}\right.
\][/tex]

A. [tex]\(\frac{x^2}{16} - \frac{y^2 - 25}{9} = 1\)[/tex]

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

C. [tex]\(\frac{x^2}{16} - \frac{25 - y^2}{9} = 1\)[/tex]

D. [tex]\(\frac{25 - y^2}{16} - \frac{y^2}{9} = 1\)[/tex]

Sagot :

GinnyAnswer
To solve the given system of equations, we can follow the logical steps to isolate [tex]\( x^2 \)[/tex] in the first equation and then substitute it into the second equation to find the resulting equation.

Given the system of equations:
[tex]\[ \left\{\begin{array}{l} x^2 + y^2 = 25 \\ \frac{x^2}{16} - \frac{y^2}{9} = 1 \end{array}\right. \][/tex]

Let's begin by isolating [tex]\( x^2 \)[/tex] in the first equation:
[tex]\[ x^2 + y^2 = 25 \][/tex]

Rearrange to solve for [tex]\( x^2 \)[/tex]:
[tex]\[ x^2 = 25 - y^2 \][/tex]

Now we substitute this expression for [tex]\( x^2 \)[/tex] into the second equation:
[tex]\[ \frac{x^2}{16} - \frac{y^2}{9} = 1 \][/tex]

Substituting [tex]\( x^2 = 25 - y^2 \)[/tex] into the second equation gives us:
[tex]\[ \frac{25 - y^2}{16} - \frac{y^2}{9} = 1 \][/tex]

This is the resulting equation after substituting [tex]\( x^2 \)[/tex] from the first equation into the second equation.

Thus, the correct choice is:
[tex]\[ \boxed{\frac{25 - y^2}{16} - \frac{y^2}{9} = 1} \][/tex]

The corresponding option from the given choices is:
[tex]\[ D. \frac{25 - y^2}{16} - \frac{y^2}{9} = 1 \][/tex]
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