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]
